2026-07-21T23:20:45Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13589 2024-03-06T10:52:23Z ddc:004 open_access

Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers Mudigonda, Abhijit S. Williams, R. Ryan Time-space tradeoffs lower bounds QMA A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the SAT problem and related problems within the polynomial-time hierarchy. For example, for the SAT problem, the state-of-the-art is that the problem cannot be solved by random-access machines in n^c time and n^o(1) space simultaneously for c < 2cos(π/7) ≈ 1.801. We extend this lower bound approach to the quantum and randomized domains. Combining Grover’s algorithm with components from SAT time-space lower bounds, we show that there are problems verifiable in O(n) time with quantum Merlin-Arthur protocols that cannot be solved in n^c time and n^o(1) space simultaneously for c < (3+√3)/2 ≈ 2.366, a super-quadratic time lower bound. This result and the prior work on SAT can both be viewed as consequences of a more general formula for time lower bounds against small-space algorithms, whose asymptotics we study in full. We also show lower bounds against randomized algorithms: there are problems verifiable in O(n) time with (classical) Merlin-Arthur protocols that cannot be solved in n^c randomized time and O(log n) space simultaneously for c < 1.465, improving a result of Diehl. For quantum Merlin-Arthur protocols, the lower bound in this setting can be improved to c < 1.5. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Abhijit S. Mudigonda and R. Ryan Williams 2021 Is Part Of LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2021.50 urn:nbn:de:0030-drops-135897 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.50 eng https://creativecommons.org/licenses/by/3.0/legalcode