A line of work initiated by Fortnow in 1997 has proven model-independent\ntime-space lower bounds for the $\\mathsf{SAT}$ problem and related problems\nwithin the polynomial-time hierarchy. For example, for the $\\mathsf{SAT}$\nproblem, the state-of-the-art is that the problem cannot be solved by\nrandom-access machines in $n^c$ time and $n^{o(1)}$ space simultaneously for $c\n< 2\\cos(\\frac{\\pi}{7}) \\approx 1.801$.\n We extend this lower bound approach to the quantum and randomized domains.\nCombining Grover's algorithm with components from $\\mathsf{SAT}$ time-space\nlower bounds, we show that there are problems verifiable in $O(n)$ time with\nquantum Merlin-Arthur protocols that cannot be solved in $n^c$ time and\n$n^{o(1)}$ space simultaneously for $c < \\frac{3+\\sqrt{3}}{2} \\approx 2.366$, a\nsuper-quadratic time lower bound. This result and the prior work on\n$\\mathsf{SAT}$ can both be viewed as consequences of a more general formula for\ntime lower bounds against small-space algorithms, whose asymptotics we study in\nfull.\n We also show lower bounds against randomized algorithms: there are problems\nverifiable in $O(n)$ time with (classical) Merlin-Arthur protocols that cannot\nbe solved in $n^c$ randomized time and $n^{o(1)}$ space simultaneously for $c <\n1.465$, improving a result of Diehl. For quantum Merlin-Arthur protocols, the\nlower bound in this setting can be improved to $c < 1.5$.\n