The Entangled Quantum Polynomial Hierarchy Collapses

TL;DR

This work defines the entangled quantum polynomial hierarchy (QEPH), allowing entangled quantum proofs across multiple rounds, and proves a striking collapse: QEPH collapses to its second level, yielding QEPH = QEΣ2 = QRG(1) ⊆ PSPACE. It further establishes that PH ⊆ QCPH ⊆ QPH and that distributions over classical proofs do not increase power, i.e., DistributionQCPH = QCPH and DistributionPH = PH. A key technical contribution is a generalized min–max framework showing QEΣ3 values reduce to QEΣ2 values, enabling collapse without expanding proof size, and a Lipton–Young-style sparsification result that reduces distribution supports to polynomial size. The findings unify and separate quantum and classical hierarchies, show robustness to error parameters, and provide a foundation for exploring the limits of quantum proof systems, with open questions about connections to broader oracle and hierarchy frameworks.

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).

Figures (1)

• Figure 1: (Color) The quantum polynomial hierarchy landscape in light of our work. The containments and complexity classes shown in grey were previously known, and the containments and complexity classes in red are contributions of this work.

Theorems & Definitions (35)

• Theorem 1.1: Combination of \ref{['lem:ent-qph-collapse', 'thm:qrg-one-equals-pspace']}
• Theorem 1.2: Informal version of \ref{['corr:entQSigmaPoly=2']}
• Theorem 1.3: Restatement of \ref{['thm:direct-proof-ph-prodqph']}
• Theorem 1.4: Restatement of \ref{['cor:distqcph=qcph']}
• Theorem 1.5: Restatement of \ref{['thm:mixedph=ph']}
• Theorem 2.2: A weaker version of Sion's min-max theorem sion1958general
• proof
• Definition 2.5: $\mathsf{\Sigma^{p}_{i}}\xspace$
• Definition 2.6: The polynomial hierarchy ($\mathsf{PH}\xspace$) stockmeyer1976polynomial
• Definition 2.7: Polynomial-time uniform family of quantum circuits