How hard is it to fake entanglement? A complexity theoretic view of nonlocality and its applications to delegating quantum computation

Is it possible to operationally distinguish every entangled state from all separable states? This is a long-standing open question in quantum information. More concretely, assuming that two non-communicating parties interact classically with a verifier, can the verifier distinguish the following two cases: (i) the parties have access to an entangled state, (ii) they have access to a separable state only (a local hidden variable model). In this work, we define a computational version of state non-locality, and show that if every entangled state exhibits such non-locality then $\mathtt{BQP} \neq \mathtt{PP}$. Surprisingly, we demonstrate how this result implies that if a one-round delegation of quantum computation (DQC) exists then $\mathtt{BQP} \neq \mathtt{PP}$. This gives a necessary complexity-theoretic assumption needed for the existence of such DQC. Our proof technique essentially builds a framework that allows one to give stronger lower-bounds for DQC by proving upper-bounds for the complexity of local-hidden-variable models.