Note from the Research Highlights Co-Chairs: A Research Highlights paper appearing in Communications is usually peer-reviewed prior to publication. The following paper is unusual in that it is still under review. However, the result has generated enormous excitement in the research community, and came strongly nominated by SIGACT, a nomination seconded by external reviewers. The complexity class NP characterizes the collection of computational problems that have efficiently verifiable solutions. With the goal of classifying computational problems that seem to lie beyond NP, starting in the 1980s complexity theorists have considered extensions of the notion of efficient verification that allow for the use of randomness (the class MA), interaction (the class IP), and the possibility to interact with multiple proofs, or provers (the class MIP). The study of these extensions led to the celebrated PCP theorem and its applications to hardness of approximation and the design of cryptographic protocols. In this work, we study a fourth modification to the notion of efficient verification that originates in the study of quantum entanglement. We prove the surprising result that every problem that is recursively enumerable, including the Halting problem, can be efficiently verified by a classical probabilistic polynomial-time verifier interacting with two all-powerful but noncommunicating provers sharing entanglement. The result resolves long-standing open problems in the foundations of quantum mechanics (Tsirelson's problem) and operator algebras (Connes' embedding problem).

We introduce a simple two-player test which certifies that the players apply tensor products of Pauli σ X and σ Z observables on the tensor product of n EPR pairs. The test has constant robustness: any strategy achieving success probability within an additive ǫ of the optimal must be poly(ǫ)-close, in the appropriate distance measure, to the honest n-qubit strategy. The test involves 2n-bit questions and 2-bit answers. The key technical ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld.As applications of our result we give (i) the first robust self-test for n EPR pairs; (ii) a quantum multiprover interactive proof system for the local Hamiltonian problem with a constant number of provers and classical questions and answers, and a constant completenesssoundness gap independent of system size; (iii) a robust protocol for delegated quantum computation.

We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability 1 in the game, or when no such strategy succeeds with probability larger than 1 2 . This proves the "games quantum PCP conjecture" of Fitzsimons and the second author (ITCS'15), albeit under randomized reductions.The core component in our reduction is a construction of a family of two-player games for testing n-qubit maximally entangled states. For any integer n ≥ 2, we give such a game in which questions from the verifier are O(log n) bits long, and answers are poly(log log n) bits long. We show that for any constant ε ≥ 0, any strategy that succeeds with probability at least 1 − ε in the test must use a state that is within distance δ(ε) = O(ε c ) from a state that is locally equivalent to a maximally entangled state on n qubits, for some universal constant c > 0. The construction is based on the classical plane-vs-point test for multivariate low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test to the quantum regime by executing independent copies of the test in the generalized Pauli X and Z bases over F q , where q is a sufficiently large prime power, and combine the two through a test for the Pauli twisted commutation relations.Our main complexity-theoretic result is obtained by combining this family of games with techniques from the classical PCP literature. More specifically, we use constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially rely on a linear property of such PCPs. Another consequence of our results is a deterministic reduction from the games quantum PCP conjecture to a suitable formulation of the constraint satisfaction quantum PCP conjecture.give two alternate formulations of Theorem 1.2 that would also establish the same QMA-hardness result, under deterministic reductions, provided that either:(i) it is QMA-hard to approximate the minimum energy of a local Hamiltonian in Y-free form (Definition 6.8) to within constant accuracy (this is a variant of the quantum PCP conjecture for local Hamiltonians), or (ii) it is QMA hard to approximate the ground energy of (not necessarily local) frustration-free Hamiltonian whose every term is a tensor product of generalized Pauli τ X or τ Z observables.Note that point (i) amounts to a deterministic reduction from Conjecture 1.2 to the constraint satisfaction quantum PCP conjecture, and establishes the first proven relation between the two conjectures (see [GKP16] for an incomparable result that relates stronger variants of both conjectures). Point (ii) is arguably a weaker assumption, as the gap is not required to be a constant and the terms of the Hamiltonian are not required to be local. However, due to the restriction that the Hamiltonian is frustration-free, it is currently not known whether the problem is QMA...

We present a stronger version of the Doherty-Parrilo-Spedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing which is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.

Semidefinite programs (SDPs) are a framework for exact or approximate optimization with widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs, meaning instances where the SDP value is far from the true optimum. These are based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory, where previously no ω(1)-round integrality gaps were known: