Two Prover Perfect Zero Knowledge for MIP

The recent MIP ˚" RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows that the complexity class MIP ˚of multiprover proof systems with entangled provers contains all recursively enumerable languages. Prior work of Grilo, Slofstra, and Yuen [FOCS '19] further shows (via a technique called simulatable codes) that every language in MIP ˚has a perfect zero knowledge (PZK) MIP ˚protocol. The MIP ˚" RE theorem uses two-prover one-round proof systems, and hence such systems are complete for MIP ˚. However, the construction in Grilo, Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect zero knowledge with two provers via simulatable codes. This leads to a natural question: are there two-prover PZK-MIP ˚protocols for all of MIP ˚?

In this paper, we show that every language in MIP ˚has a two-prover oneround PZK-MIP ˚protocol, answering the question in the affirmative. For the proof, we use a new method based on a key consequence of the MIP ˚" RE theorem, which is that every MIP ˚protocol can be turned into a family of boolean constraint system (BCS) nonlocal games. This makes it possible to work with MIP ˚protocols as boolean constraint systems, and in particular allows us to use a variant of a construction due to Dwork, Feige, Kilian, Naor, and Safra [Crypto '92] which gives a classical MIP protocol for 3SAT with perfect zero knowledge. To show quantum soundness of this classical construction, we develop a toolkit for analyzing quantum soundness of reductions between BCS games, which we expect to be useful more broadly. This toolkit also applies to commuting operator strategies, and our argument shows that every language with a commuting operator BCS protocol has a two prover PZK commuting operator protocol.