Let $R_\epsilon(\cdot)$ stand for the bounded-error randomized query complexity with error $\epsilon > 0$. For any relation $f \subseteq \{0,1\}^n \times S$ and partial Boolean function $g \subseteq \{0,1\}^m \times \{0,1\}$, we show that $R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \sqrt{R_{1/3}(g)})$, where $f \circ g^n \subseteq (\{0,1\}^m)^n \times S$ is the composition of $f$ and $g$. We give an example of a relation $f$ and partial Boolean function $g$ for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity $\bar \chi(g)$ of a partial Boolean function $g$. We show $\bar \chi(g) \in \Omega(\sqrt{R_{1/3}(g)})$ for any (partial) function $g$ and $R_{1/3}(f \circ g^n) \in \Omega(R_{4/9}(f) \cdot \bar \chi(g))$; these two bounds imply our composition result. We further show that $\bar \chi(g)$ is always at least as large as the sabotage complexity of $g$, introduced by Ben-David and Kothari.

In communication complexity the Arthur-Merlin (AM) model is the most natural one that allows both randomness and nondeterminism. Presently we do not have any superlogarithmic lower bound for the AM-complexity of an explicit function. Obtaining such a bound is a fundamental challenge to our understanding of communication phenomena. In this article we explore the gap between the known techniques and the complexity class AM. In the first part we define a new natural class, Small-advantage Layered Arthur-Merlin (SLAM), that has the following properties: SLAM is (strictly) included in AM and includes all previously known subclasses of AM with non-trivial lower bounds: NP; MA; SBP; UAM subset of SLAM subset of AM . Note that NP subset of MA subset of SBP, while SBP and UAM are known to be incomparable. SLAM is qualitatively stronger than the union of those classes: f is an element of SLAM \ (SBP boolean OR UAM) holds for an (explicit) partial function f. SLAM is a subject to the discrepancy bound: for any f SLAM (f) is an element of Omega (root log 1/disc(f)) In particular, the inner product function does not have an efficient SLAM-protocol. Structurally this can be summarised as SBP boolean OR UAM subset of SLAM subset of AM \ PP. In the second part we ask why proving a lower bound of w(root n) on the MA-complexity of an explicit function seems to be difficult. We show that such a bound either must explore certain "uniformity" of MA (which would require a rather unusual argument), or would imply a non-trivial lower bound on the AM-complexity of the same function. Both of these results are related to the notion of layer complexity, which is, informally, the number of "layers of nondeterminism" used by a protocol.

The notion of semi-random sources , also known as Santha-Vazirani ( SV ) sources, stands for a sequence of n bits, where the dependence of the i’th bit on the previous i − 1 bits is limited for every i ∈ [ n ]. If the dependence of the i’th bit on the remaining n − 1 bits is limited, then this is a strong SV - source . Even the strong SV -sources are known not to admit (universal) deterministic extractors , but they have seeded extractors , as their min-entropy is Ω n. It is intuitively obvious that strong SV -sources are more than just high-min-entropy sources , and this work explores the intuition. Deterministic condensers are known not to exist for general high-min-entropy sources, and we construct for any constants ε , δ ∈ (0,1) a deterministic condenser that maps n bits coming from a strong SV -source with bias at most δ to Ω n bits of min-entropy rate at least 1 − ε . In conclusion we observe that deterministic condensers are closely related to very strong extractors – a proposed strengthening of the notion of strong (seeded) extractors : in particular, our constructions can be viewed as very strong extractors for the family of strong Santha-Vazirani distributions . The notion of very strong extractors requires that the output remains unpredictable even to someone who knows not only the seed value (as in the case of strong extractors), but also the extractor’s outputs corresponding to the same input value with each of the preceding seed values (say, under the lexicographic ordering). Very strong extractors closely resemble the original notion of SV -sources, except that the bits must satisfy the unpredictability requirement only on average.

One of the major open problems in complexity theory is proving superlogarithmic lower bounds on the depth of circuits (i.e., $\mathbf{P}\not\subseteq\mathbf{NC}^1$). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving superpolynomial circuit lower bounds. Karchmer, Raz, and Wigderson [Comput. Complexity, 5 (1995), pp. 191--204] suggested approaching this problem by proving the following conjecture: given two Boolean functions $f$ and $g$, the depth complexity of the composed function $g\diamond f$ is roughly the sum of the depth complexities of $f$ and $g$. They showed that the validity of this conjecture would imply that $\mathbf{P}\not\subseteq\mathbf{NC}^1$. As a starting point for studying the composition of functions, they introduced a relation called “the universal relation” and suggested studying the composition of universal relations. This suggestion proved fruitful, and an analogue of the Karchmer--Raz--Wigderson (KRW) conjecture for the universal relation was proved by Edmonds et al. [Comput. Complexity, 10 (2001), pp. 210--246]. An alternative proof was given later by H\aastad and Wigderson [in Advances in Computational Complexity Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 13, AMS, Providence, RI, 1993, pp. 119--134]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still an open question. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation.

We introduce a new concept, which we call partition expanders. The basic idea is to study quantitative properties of graphs in a slightly different way than it is in the standard definition of expanders. While in the definition of expanders it is required that the number of edges between any pair of sufficiently large sets is close to the expected number, we consider partitions and require this condition only for most of the pairs of blocks. As a result, the blocks can be substantially smaller. We show that for some range of parameters, to be a partition expander a random graph needs exponentially smaller degree than any expander would require in order to achieve similar expanding properties. We apply the concept of partition expanders in communication complexity. First, we construct an optimal pseudo-random generator (PRG) for the Simultaneous Message Passing (SMP) model: it needs n + log k random bits against protocols of cost Ω(k). Second, we compare the SMP model to that of Simultaneous Two-Way Communication, and give a new separation that is stronger both qualitatively and quantitatively than the previously known ones.

How low can the joint entropy of $n$ $d$-wise independent (for $d\ge2$) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$, for $p<1$)? This question has been posed and partially answered in a recent work of Babai. In this paper we improve some of his bounds, prove new bounds in a wider range of parameters and show matching upper bounds in some special cases. In particular, we prove tight lower bounds for the min-entropy (as well as the entropy) of pairwise and three-wise independent balanced binary variables for infinitely many values of $n$.

We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions. We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information $k$, showing that it is $\Theta(\sqrt{n(k+1)})$ for all $0\leq k\leq n$. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case ($k=0$), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs $\Omega(n)$ bits of mutual information in the corresponding distribution. We study the same question in the distributional quantum setting, and show a lower bound of $\Omega((n(k+1))^{1/4})$, and an upper bound, matching up to a logarithmic factor. We show that there are total Boolean functions $f_d$ on $2n$ inputs that have distributional communication complexity $O(\log n)$ under all distributions of information up to $o(n)$, while the (interactive) distributional complexity maximised over all distributions is $\Theta(\log d)$ for $6n\leq d\leq 2^{n/100}$. We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error $\epsilon$ is multiplicative in $\log(1/\epsilon)$ -- the previous upper bounds had the dependence of more than $1/\epsilon$.

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,...,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,...,Y_r$.

We develop a new lower bound method for analysing the complexity of the Equality function (EQ) in the Simultaneous Message Passing (SMP) model of communication complexity. The new technique gives tight lower bounds of $\Omega(\sqrt n)$ for both EQ and its negation NE in the non-deterministic version of quantum-classical SMP, where Merlin is also quantum $-$ this is the strongest known version of SMP where the complexity of both EQ and NE remain high (previously known techniques seem to be insufficient for this). Besides, our analysis provides to a unified view of the communication complexity of EQ and NE, allowing to obtain tight characterisation in all previously studied and a few newly introduced versions of SMP, including all possible combination of either quantum or randomised Alice, Bob and Merlin in the non-deterministic case. Some of our results highlight that NE is easier than EQ in the presence of classical proofs, whereas the problems have (roughly) the same complexity when a quantum proof is present.

Two parties wish to carry out certain distributed computational tasks, and they are given access to a source of correlated random bits. It allows the parties to act in a correlated manner, which can be quite useful. But what happens if the shared randomness is not perfect? In this work, we initiate the study of the power of different sources of shared randomness in communication complexity. This is done in the setting of simultaneous message passing (SMP) model of communication complexity, which is one of the most suitable models for studying the resource of shared randomness. Toward characterising the power of various sources of shared randomness, we introduce a measure for the quality of a source - we call it collision complexity. Our results show that the collision complexity tightly characterises the power of a (shared) randomness resource in the SMP model. Of independent interest is our demonstration that even the weakest sources of shared randomness can in some cases increase the power of SMP substantially: the equality function can be solved very efficiently with virtually any nontrivial shared randomness.

One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P ⊈ NC 1 ). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds. Karchmer, Raz, and Wigderson [21] suggested to approach this problem by proving the following conjecture: given two boolean functions f and g , the depth complexity of the composed function g o f is roughly the sum of the depth complexities of f and g . They showed that the validity of this conjecture would imply that P ⊈ NC 1 . As a starting point for studying the composition of functions, they introduced a relation called "the universal relation", and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [12]. An alternative proof was given later by Håstad and Wigderson [18]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it. Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations -- communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.

We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that low-rank matrices have efficient protocols in any of the aforementioned measures. Furthermore, we show that the notion of zero-communication complexity is equivalent to an extension of the common discrepancy bound. Linial et al. [Combinatorica, 2007] showed that the discrepancy of a sign matrix is lower-bounded by an inverse polynomial in the logarithm of the associated matrix. We show that if these results can be generalized to the extended discrepancy, this will imply the log-rank conjecture.

We introduce a new type of cryptographic primitive that we call hiding fingerprinting. A (quantum) fingerprinting scheme translates a binary string of length $n$ to $d$ (qu)bits, typically $d\ll n$, such that given any string $y$ and a fingerprint of $x$, one can decide with high accuracy whether $x=y$. Classical fingerprinting schemes cannot hide information very well: a classical fingerprint of $x$ that guarantees error at most $\epsilon$ necessarily reveals $\Omega(\log(1/ epsilon))$ bits about $x$. We call a scheme hiding if it reveals $o(\log(1/\epsilon))$ bits; accordingly, no classical scheme is hiding. For any constant $c$, we construct two kinds of hiding fingerprinting schemes, both mapping $n$-bit strings to $O(\log n)$ qubits and guaranteeing one-sided error probability at most $1/n^c$. The first kind uses pure states and leaks at most O(1) bits, and the second kind uses mixed states and leaks at most $1/n^c$ bits, where the "leakage" is bounded via accessible information. The schemes are computationally efficient. Our mixed-state scheme is optimal, as shown via a generic strategy that extracts $1/\poly(n)$ bits from any fingerprint over $O(\log n)$ qubits.

We study shared randomness in the context of multi-party number-in-hand communication protocols in the simultaneous message passing model. We show that with three or more players, shared randomness exhibits new interesting properties that have no direct analogues in the two-party case.First, we demonstrate a hierarchy of modes of shared randomness, with the usual shared randomness where all parties access the same random string as the strongest form in the hierarchy. We show exponential separations between its levels, and some of our bounds may be of independent interest. For example, we show that the equality function can be solved by a protocol of constant length using the weakest form of shared randomness, which we call XOR-shared randomness.Second, we show that quantum communication cannot replace shared randomness in the k-party case, where k >= 3 is any constant. We demonstrate a promise function GP(k) that can be computed by a classical protocol of constant length when (the strongest form of) shared randomness is available, but any quantum protocol without shared randomness must send n(Omega(1)) qubits to compute it. Moreover, the quantum complexity of GP(k) remains n(Omega(1)) even if the "second strongest" mode of shared randomness is available. While a somewhat similar separation was already known in the two-party case, in the multi-party case our statement is qualitatively stronger:In the two-party case, only a relational communication problem with similar properties is known.In the two-party case, the gap between the two complexities of a problem can be at most exponential, as it is known that 2(O(c)) log n qubits can always replace shared randomness in any c-bit protocol. Our bounds imply that with quantum communication alone, in general, it is not possible to simulate efficiently even a three-bit three-party classical protocol that uses shared randomness.

We consider the problem of constructing pseudorandom generators for read-once circuits. We give an explicit construction of a pseudorandom generator for the class of read-once constant depth circuits with unbounded fan-in AND, OR, NOT and generalized modulo m gates, where m is an arbitrary fixed constant. The seed length of our generator is poly-logarithmic in the number of variables and the error.

We construct a new quantum algorithm for the graph collision problem; that is, the problem of deciding whether the set of marked vertices contains a pair of adjacent vertices in a known graph G. The query complexity of our algorithm is O(sqrt(n)+sqrt(alpha*(G))), where n is the number of vertices and alpha*(G) is the maximum total degree of the vertices in an independent set of G. Notably, if G is a random graph where every edge is present with a fixed probability independently of other edges, then our algorithm requires O(sqrt(n log n)) queries on most graphs, which is optimal up to the sqrt(log n) factor on most graphs.

We define a new model of quantum learning that we call Predictive Quantum (PQ). This is a quantum analogue of PAC, where during the testing phase the student is only required to answer a polynomial number of testing queries. We demonstrate a relational concept class that is efficiently learnable in PQ, while in any reasonable classical model exponential amount of training data would be required. This is the first unconditional separation between quantum and classical learning. We show that our separation is the best possible in several ways; in particular, there is no analogous result for a functional class, as well as for several weaker versions of quantum learning. In order to demonstrate tightness of our separation we consider a special case of one-way communication that we call single-input mode, where Bob receives no input. Somewhat surprisingly, this setting becomes nontrivial when relational communication tasks are considered. In particular, any problem with two-sided input can be transformed into a single-input relational problem of equal classical one-way cost. We show that the situation is different in the quantum case, where the same transformation can make the communication complexity exponentially larger. This happens if and only if the original problem has exponential gap between quantum and classical one-way communication costs. We believe that these auxiliary results might be of independent interest.

We propose and construct a quantum money scheme that allows verification through classical communication with a bank. This is the first demonstration that a secure quantum money scheme exists that does not require quantum communication for coin verification. Our scheme is secure against adaptive adversaries -- this property is not directly related to the possibility of classical verification, nevertheless none of the earlier quantum money constructions is known to possess it.