Quantum free games

The complexity of free games with two or more classical players was essentially settled by Aaronson, Impagliazzo, and Moshkovitz (CCC'14). There are two complexity classes that can be considered quantum analogues of classical free games: (1) AM*, the multiprover interactive proof class corresponding to free games with entangled players, and, somewhat less obviously, (2) BellQMA(2), the class of quantum Merlin-Arthur proof systems with two unentangled Merlins, whose proof states are separately measured by Arthur. In this work, we make significant progress towards a tight characterization of both of these classes. 1. We show a BellQMA(2) protocol for 3SAT on $n$ variables, where the total amount of communication is $\tilde{O}(\sqrt{n})$. This answers an open question of Chen and Drucker (2010) and also shows, conditional on ETH, that the algorithm of Brand\~{a}o, Christandl and Yard (STOC'11) is tight up to logarithmic factors. 2. We show that $\mathsf{AM}^*[n_{\text{provers}} = 2, q = O(1), a =\mathrm{poly}\log(n)] = \mathsf{RE}$, i.e. that free entangled games with constant-sized questions are as powerful as general entangled games. Our result is a significant improvement over the headline result of Ji et al. (2020), whose MIP* protocol for the halting problem has $\mathrm{poly}(n)$-sized questions and answers. 3. We obtain a zero-gap AM* protocol for a $\Pi_2$ complete language with constant-size questions and almost logarithmically large answers, improving on the headline result of Mousavi, Nezhadi and Yuen (STOC'22). 4. Using a connection to the nonuniform complexity of the halting problem we show that any MIP* protocol for RE requires $\Omega(\log n)$ bits of communication. It follows that our results in item 3 are optimal up to an $O(\log^* n)$ factor, and that the gapless compression theorems of MNY'22 are asymptotically optimal.