10.4230/LIPIcs.MFCS.2018.58
Gharibian, Sevag
Sevag
Gharibian0000-0002-9992-3379
Santha, Miklos
Miklos
Santha
Sikora, Jamie
Jamie
Sikora
Sundaram, Aarthi
Aarthi
Sundaram
Yirka, Justin
Justin
Yirka
Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)
Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
2018
2019-03-01T11:29:44Z
2019-03-06T22:11:03Z
conference
https://ris.uni-paderborn.de/record/8161
https://ris.uni-paderborn.de/record/8161.json
1805.11139
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, Q Sigma_3, 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 P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP.