A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form F(x,y) = f(x_1 y_1, …, x_n y_n), when the outer function f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors. In this work, we settle this problem in a strong way. We show that for every Boolean function f, the bounded-error quantum and classical deterministic communication complexities of the function f ∘AND_2 are polynomially related, up to polylogarithmic factors in n. We prove this by showing that both are characterized–up to polynomial loss–by the logarithm of the De Morgan sparsity of f. Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.
We construct a universal decompressor U for plain Kolmogorov complexity C_U such that the Halting Problem cannot be decided by any polynomial-time oracle machine with access to the set of random strings R_C_U = {x : C_U(x) ≥ |x|}. This result resolves a problem posed by Eric Allender regarding the computational power of Kolmogorov complexity-based oracles.
We study sparse polynomials with bounded individual degree and their factors, obtaining the following structural and algorithmic results. 1. A deterministic polynomial-time algorithm to find all sparse divisors of a sparse polynomial of bounded individual degree, together with the first upper bound on the number of non-monomial irreducible factors of such polynomials. 2. A poly(n,s^dlogℓ)-time algorithm that recovers ℓ irreducible s-sparse polynomials of individual degree at most d from blackbox access to their (not necessarily sparse) product. This partially resolves a question of Dutta-Sinhababu-Thierauf (RANDOM 2024). In particular, if ℓ=O(1) the algorithm runs in polynomial time. 3. Deterministic algorithms for factoring a product of s-sparse polynomials of individual degree d from blackbox access. Over fields of characteristic zero or sufficiently large characteristic the runtime is poly(n,s^d^3log n); over arbitrary fields it is poly(n,(d^2)!,s^d^5log n). This improves Bhargava-Saraf-Volkovich (JACM 2020), which gives poly(n,s^d^7log n) time for a single sparse polynomial. For a single sparse input we obtain poly(n,s^d^2log n) time. 4. Given blackbox access to a product of factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm to find all irreducible sparse multiquadratic factors with multiplicities. This generalizes the algorithms of Volkovich (RANDOM 2015, 2017) and extends the complete-power test of Bisht-Volkovich (CC 2025). To handle arbitrary fields we introduce a notion of primitive divisors that removes characteristic assumptions from most of our algorithms.
Zero-knowledge codes, introduced by Decatur, Goldreich, and Ron (ePrint 1997), are error-correcting codes in which few codeword symbols reveal no information about the encoded message, and have been extensively used in cryptographic constructions. Quantum CSS codes, introduced by Calderbank and Shor (Phys. Rev. A 1996) and Steane (Royal Society A 1996), are error-correcting codes that allow for quantum error correction, and are also useful for applications in quantum complexity theory. In this short note, we show that (linear, perfect) zero-knowledge codes and quantum CSS codes are equivalent. We demonstrate the potential of this equivalence by using it to obtain explicit asymptotically-good zero-knowledge locally-testable codes.
A symbolic determinant under rank-one restriction computes a polynomial of the form $\det(A_0+A_1y_1+\ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are square matrices over a field $\mathbb{F}$ and $rank(A_i)=1$ for each $i\in[n]$. This class of polynomials has been studied extensively, since the work of Edmonds (1967), in the context of linear matroids, matching, matrix completion and polynomial identity testing. We study the following learning problem for this class: Given black-box access to an $n$-variate polynomial $f=\det(A_0+A_1y_1+ \ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are unknown square matrices over $\mathbb{F}$ and rank$(A_i)=1$ for each $i\in[n]$, find a square matrix $B_0$ and rank-one square matrices $B_1,\ldots,B_n$ over $\mathbb{F}$ such that $f=\det(B_0+B_1y_1+\ldots+B_ny_n)$. In this work, we give a randomized poly(n) time algorithm to solve this problem. As the above-mentioned class is known to be equivalent to the class of read-once determinants (RODs), we will refer to the problem as learning RODs. The algorithm for learning RODs is obtained by connecting with a well-known open problem in linear algebra, namely the Principal Minor Assignment Problem (PMAP), which asks to find (if possible) a matrix having prescribed principal minors. PMAP has also been studied in machine learning to learn the kernel matrix of a determinantal point process. Here, we study a natural black-box version of PMAP: Given black-box access to an $n$-variate polynomial $f = \det(A + Y)$, where $A \in \mathbb{F}^{n \times n}$ is unknown and $Y = diag(y_1,\ldots,y_n)$, find a $B\in\mathbb{F}^{n\times n}$ such that $f=det(B+Y)$. We show that black-box PMAP can be solved in randomized poly(n) time, and further, it is randomized polynomial-time equivalent to learning RODs. We resolve black-box PMAP by investigating a property of dense matrices that we call the rank-one extension property.
Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a century ago, is to encode the data into a redundant form, so that the original information can be decoded from the redundant encoding even in the presence of some noise or corruption. One prominent family of error-correcting codes are Reed-Solomon Codes which encode the data using evaluations of low-degree polynomials. Nearly six decades after they were introduced, Reed-Solomon Codes, as well as some related families of polynomial-based codes, continue to be widely studied, both from a theoretical perspective and from the point of view of applications. Besides their obvious use in communication, error-correcting codes such as Reed-Solomon Codes are also useful for various applications in theoretical computer science. These applications often require the ability to cope with many errors, much more than what is possible information-theoretically. List-decodable codes are a special class of error-correcting codes that enable correction from more errors than is traditionally possible by allowing a small list of candidate decodings. These codes have turned out to be extremely useful in various applications across theoretical computer science and coding theory. In recent years, there have been significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes. This includes efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms. In this book, we survey these developments.
In this work, we establish the first separation between computation with bounded and unbounded space, for problems with short outputs (i.e., working memory can be exponentially larger than output size), both in the classical and the quantum setting. Towards that, we introduce a problem called nested collision finding, and show that optimal query complexity can not be achieved without exponential memory. Our result is based on a novel “two-oracle recording” technique, where one oracle “records” the computation's long outputs under the other oracle, effectively reducing the time-space trade-off for short-output problems to that of long-output problems. We believe this technique will be of independent interest for establishing time-space tradeoffs in other short-output settings.
We introduce spiky rank, a new matrix parameter that enhances blocky rank by combining the combinatorial structure of the latter with linear-algebraic flexibility. A spiky matrix is block-structured with diagonal blocks that are arbitrary rank-one matrices, and the spiky rank of a matrix is the minimum number of such matrices required to express it as a sum. This measure extends blocky rank to real matrices and is more robust for problems with both combinatorial and algebraic character. Our conceptual contribution is as follows: we propose spiky rank as a well-behaved candidate matrix complexity measure and demonstrate its potential through applications. We show that large spiky rank implies high matrix rigidity, and that spiky rank lower bounds yield lower bounds for depth-2 ReLU circuits, the basic building blocks of neural networks. On the technical side, we establish tight bounds for random matrices and develop a framework for explicit lower bounds, applying it to Hamming distance matrices and spectral expanders. Finally, we relate spiky rank to other matrix parameters, including blocky rank, sparsity, and the γ_2-norm.
We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-4 arithmetic circuits of the form Σ^[r]∧^[d]Σ^[s]Π^[δ]. This model generalizes Waring decompositions and diagonal circuits, and captures sums of powers of low-degree sparse polynomials. Specifically, each circuit computes a sum of r terms, where each term is a d-th power of an s-sparse polynomial of degree δ. This model also includes algebraic representations that arise in tensor decomposition and moment-based learning tasks such as mixture models and subspace learning. We give deterministic worst-case algorithms for PIT and reconstruction in this model. Our PIT construction applies when d>r^2 and yields explicit hitting sets of size O(r^4 s^4 n^2 d δ^3). The reconstruction algorithm runs in time poly(n,s,d) under the condition d=Ω(r^4δ), and in particular it tolerates polynomially large top fan-in r and bottom degree δ. Both results hold over fields of characteristic zero and over fields of sufficiently large characteristic. These algorithms provide the first polynomial-time deterministic solutions for depth-4 powering circuits with unbounded top fan-in. In particular, the reconstruction result improves upon previous work which required non-degeneracy or average-case assumptions. The PIT construction relies on the ABC theorem for function fields (Mason-Stothers theorem), which ensures linear independence of high-degree powers of sparse polynomials after a suitable projection. The reconstruction algorithm combines this with Wronskian-based differential operators, structural properties of their kernels, and a robust version of the Klivans-Spielman hitting set.
We show that Hilbert's Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that the number of solutions to a system of equations can be computed in polynomial time with oracle access to the counting hierarchy. Our results hold in particular for polynomials with coefficients in either the rational numbers or a finite field. Previously, the best-known bounds on the complexities of these problems were PSPACE and FPSPACE, respectively. Our main technical contribution is the construction of a uniform family of constant-depth arithmetic circuits that compute the multivariate resultant.
