



01662nas a2200145   4500008004100000245008400041210006900125520118800194100002101382700001901403700001801422700002101440700001801461856003701479       2018                        eng d00aQuantum generalizations of the polynomial hierarchy with applications to QMA(2)0 aQuantum generalizations of the polynomial hierarchy with applica3 a<p>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&#39;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 &quot;equivalent&quot;), 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 &quot;unentanglement&quot;), QMA(2) is strictly contained in NEXP.</p>1 aGharibian, Sevag1 aSantha, Miklos1 aSikora, Jamie1 aSundaram, Aarthi1 aYirka, Justin  uhttps://arxiv.org/abs/1805.11139