Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds

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 \(\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}\).