# Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

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 n^o(1) 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.

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