TY - CONF
AB - 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.
AU - Gharibian, Sevag
AU - Santha, Miklos
AU - Sikora, Jamie
AU - Sundaram, Aarthi
AU - Yirka, Justin
ED - Potapov, Igor
ED - Spirakis, Paul
ED - Worrell, James
ID - 8161
KW - Complexity Theory
KW - Quantum Computing
KW - Polynomial Hierarchy
KW - Semidefinite Programming
KW - QMA(2)
KW - Quantum Complexity
T2 - 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
TI - Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)
VL - 117
ER -