We introduce the entangled quantum polynomial hierarchy $\mathsf{QEPH}$ as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove $\mathsf{QEPH}$ collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. As a consequence, $\mathsf{QEPH} = \mathsf{QRG(1)}$, the class of problems having one-turn quantum refereed games, which is known to be contained in $\mathsf{PSPACE}$. This is in contrast to the unentangled quantum polynomial hierarchy $\mathsf{QPH}$, which contains $\mathsf{QMA(2)}$. We also introduce a generalization of the quantum-classical polynomial hierarchy $\mathsf{QCPH}$ where the provers send probability distributions over strings (instead of strings) and denote it by $\mathsf{DistributionQCPH}$. Conceptually, this class is intermediate between $\mathsf{QCPH}$ and $\mathsf{QPH}$. We prove $\mathsf{DistributionQCPH} = \mathsf{QCPH}$, suggesting that only quantum superposition (not classical probability) increases the computational power of these hierarchies. To prove this equality, we generalize a game-theoretic result of Lipton and Young (1994) which says that the provers can send distributions that are uniform over a polynomial-size support. We also prove the analogous result for the polynomial hierarchy, i.e., $\mathsf{DistributionPH} = \mathsf{PH}$. These results also rule out certain approaches for showing $\mathsf{QPH}$ collapses. Finally, we show that $\mathsf{PH}$ and $\mathsf{QCPH}$ are contained in $\mathsf{QPH}$, resolving an open question of Gharibian et al. (2022).

翻译：我们引入纠缠量子多项式层次结构 $\mathsf{QEPH}$，作为一类可通过交替量子证明（可能相互纠缠）高效验证的问题。我们证明 $\mathsf{QEPH}$ 坍缩至其第二层。实际上，我们表明多项式数量的交替坍缩至仅两层。由此可得 $\mathsf{QEPH} = \mathsf{QRG(1)}$，即具有单回合量子验证博弈的问题类，已知其包含于 $\mathsf{PSPACE}$ 中。这不同于非纠缠量子多项式层次结构 $\mathsf{QPH}$（后者包含 $\mathsf{QMA(2)}$）。此外，我们引入量子-经典多项式层次结构 $\mathsf{QCPH}$ 的推广形式，其中证明者发送字符串上的概率分布（而非字符串本身），并将其记为 $\mathsf{DistributionQCPH}$。概念上，此类介于 $\mathsf{QCPH}$ 与 $\mathsf{QPH}$ 之间。我们证明 $\mathsf{DistributionQCPH} = \mathsf{QCPH}$，表明仅量子叠加（而非经典概率）能增强这些层次结构的计算能力。为证明该等式，我们推广了 Lipton 与 Young (1994) 的博弈论结论：证明者可发送均匀分布于多项式规模支撑集上的分布。我们还证明了多项式层次结构的类似结论，即 $\mathsf{DistributionPH} = \mathsf{PH}$。这些结果亦排除了某些证明 $\mathsf{QPH}$ 坍缩的方法。最后，我们证明 $\mathsf{PH}$ 与 $\mathsf{QCPH}$ 均包含于 $\mathsf{QPH}$ 中，解决了 Gharibian 等人 (2022) 提出的一个开放问题。