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

Abstract

A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the $\\mathsf{SAT}$ problem and related problems within the polynomial-time hierarchy. For example, for the $\\mathsf{SAT}$ 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 $c < 2\\cos(\\frac{\\pi}{7}) \\approx 1.801$. \nWe extend this lower bound approach to the quantum and randomized domains. Combining Grover s algorithm with components from $\\mathsf{SAT}$ 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 $c < \\frac{3+\\sqrt{3}}{2} \\approx 2.366$, a super-quadratic time lower bound. This result and the prior work on $\\mathsf{SAT}$ 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. \nWe 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 $c < 1.5$.