Quantum generalizations of the polynomial hierarchy with applications to QMA(2)

Abstract

The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, \(\rm{QCPH}\), uses classical proofs, and the second, \(\rm{QPH}\), uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, \(\rm{Q\Sigma_3}\), into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for \(\rm{QMA(2)}\), the variant of Quantum Merlin-Arthur (\(\rm{QMA}\)) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if \(\rm{QCPH = QPH}\) (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs ``equivalent''), then QMA(2) is in the counting hierarchy (specifically, in \({\rm P}^{{\rm pp}^{{\rm pp}}}\)). Second, because \(\rm{QMA(2)}\subseteq \rm{Q\Sigma_3}\), \(\rm{QMA(2)}\) is strictly contained in NEXP unless \(\rm{QMA(2)}=\rm{Q\Sigma_3}\) (i.e., alternating quantifiers do not help in the presence of ``unentanglement'').