The Polynomial-Time Hierarchy ($\mathsf{PH}$) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers. Quantumly, however, despite the fact that at least \emph{four} definitions of quantum $\mathsf{PH}$ exist, it has been challenging to prove analogues for these of even basic facts from $\mathsf{PH}$. This work studies three quantum-verifier based generalizations of $\mathsf{PH}$, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings ($\mathsf{QCPH}$) and quantum mixed states ($\mathsf{QPH}$) as proofs, and one of which is new to this work, utilizing quantum pure states ($\mathsf{pureQPH}$) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for $\mathsf{QCPH}$. Then, for our new class $\mathsf{pureQPH}$, we show one-sided error reduction for $\mathsf{pureQPH}$, as well as the first bounds relating these quantum variants of $\mathsf{PH}$, namely $\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}$.

翻译：多项式时间层级（$\mathsf{PH}$）是经典复杂性理论的核心概念，其应用涵盖随机化计算、电路下界以及近时期量子计算机的“量子优势”分析。然而在量子领域中，尽管至少存在*四种*量子$\mathsf{PH}$的定义，但要证明这些定义在$\mathsf{PH}$基本事实上的类似结论仍具挑战性。本文研究了基于量子验证者的三种$\mathsf{PH}$推广形式：其中两种源自[Gharibian, Santha, Sikora, Sundaram, Yirka, 2022]，分别使用经典字符串（$\mathsf{QCPH}$）和量子混合态（$\mathsf{QPH}$）作为证明；第三种为本研究新提出的，采用量子纯态（$\mathsf{pureQPH}$）作为证明。我们首先解决了[GSSSY22]中的若干开放问题，包括$\mathsf{QCPH}$的坍塌定理和Karp-Lipton定理。随后针对新类$\mathsf{pureQPH}$，我们展示了其单边误差降低性质，并建立了这些量子$\mathsf{PH}$变种间的首个包含关系，即$\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}$。