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.

翻译：具有两个或多个经典玩家的自由游戏的复杂性基本上由Aaronson、Impagliazzo和Moshkovitz（CCC'14）解决。存在两个可视为经典自由游戏量子模拟的复杂性类：（1）AM*，对应于具有纠缠玩家的自由游戏的多证明者交互证明类；（2）BellQMA(2)，即具有两个非纠缠Merlin的量子Merlin-Arthur证明系统类，其中Arthur分别测量其证明状态。在这项工作中，我们朝着对这两个类的严格刻画取得了重大进展。1. 我们展示了针对$n$变量3SAT问题的BellQMA(2)协议，其中通信总量为$\tilde{O}(\sqrt{n})$。这回答了Chen和Drucker（2010）的一个开放问题，同时表明在ETH条件下，Brandão、Christandl和Yard（STOC'11）的算法在对数因子意义下是紧的。2. 我们证明了$\mathsf{AM}^*[n_{\text{证明者}} = 2, q = O(1), a =\mathrm{poly}\log(n)] = \mathsf{RE}$，即具有常数大小问题的自由纠缠游戏与一般纠缠游戏具有相同能力。这一结果显著改进了Ji等人（2020）的主要成果，其用于停机问题的MIP*协议具有$\mathrm{poly}(n)$大小的问题与答案。3. 我们获得了针对$\Pi_2$完全语言的零间隙AM*协议，该协议具有常数大小问题和几乎对数大小的答案，改进了Mousavi、Nezhadi和Yuen（STOC'22）的主要成果。4. 利用与停机问题的非均匀复杂性的联系，我们证明任何针对RE的MIP*协议需要$\Omega(\log n)$比特的通信。由此得出，我们第3项中的结果在$O(\log^* n)$因子意义下是最优的，且MNY'22的无间隙压缩定理是渐近最优的。