In this work we consider the interplay between multiprover interactive proofs, quantum entanglement, and zero knowledge proofs - notions that are central pillars of complexity theory, quantum information and cryptography. In particular, we study the relationship between the complexity class MIP @math , the set of languages decidable by multiprover interactive proofs with quantumly entangled provers, and the class PZKMIP @math , which is the set of languages decidable by MIP @math protocols that furthermore possess the perfect zero knowledge property. Our main result is that the two classes are equal, i.e., MIP @math PZKMIP @math . This result provides a quantum analogue of the celebrated result of Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP @math PZKMIP (in other words, all classical multiprover interactive protocols can be made zero knowledge). We prove our result by showing that every MIP @math protocol can be efficiently transformed into an equivalent zero knowledge MIP @math protocol in a manner that preserves the completeness-soundness gap. Combining our transformation with previous results by Slofstra (Forum of Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we obtain the corollary that all co-recursively enumerable languages (which include undecidable problems as well as all decidable problems) have zero knowledge MIP @math protocols with vanishing promise gap.
We show that any language in nondeterministic time @math , where the number of iterated exponentials is an arbitrary function @math , can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness @math and soundness @math , where the number of iterated exponentials is @math and @math is a universal constant. The result was previously known for @math and @math ; we obtain it for any time-constructible function @math . The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC'17). As a separate consequence of this technique we obtain a different proof of Slofstra's recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson's problem on the relation between the commuting operator and tensor product models for quantum correlations. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis.