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 does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, 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, QΣ3, into NEXP {using the Ellipsoid Method for efficiently solving semidefinite programs}. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if 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 PPPPP). Second, unless QMA(2)=QΣ3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP.

1 aGharibian, Sevag1 aSantha, Miklos1 aSikora, Jamie1 aSundaram, Aarthi1 aYirka, Justin uhttps://arxiv.org/abs/1805.1113901206nas a2200169 4500008004100000245003600041210003500077260001500112300001200127490000700139520077800146100002100924700001700945700002100962700001800983856003501001 2015 eng d00aTensor network non-zero testing0 aTensor network nonzero testing c2015/07/01 a885-8990 v153 aTensor networks are a central tool in condensed matter physics. In this paper, we initiate the study of tensor network non-zero testing (TNZ): Given a tensor network T, does T represent a non-zero vector? We show that TNZ is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of TNZ on non-negative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MA-complete stoquastic k-SAT problem on D-dimensional qudits is in NP for logarithmic k and constant D. This reveals the first class of quantum Hamiltonians whose commuting variant is known to be in NP for all (1) logarithmic k, (2) constant D, and (3) for arbitrary interaction graphs.1 aGharibian, Sevag1 aLandau, Zeph1 aShin, Seung, Woo1 aWang, Guoming uhttp://arxiv.org/abs/1406.5279