The Entangled Quantum Polynomial Hierarchy Collapses

Abstract

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).