Evolutions of local Hamiltonians in short times are expected to remain local and thus limited. In this paper, we validate this intuition by proving some limitations on short-time evolutions of local time-dependent Hamiltonians. We show that the distribution of the measurement output of short-time (at most logarithmic) evolutions of local Hamiltonians are concentrated and satisfy an isoperimetric inequality. To showcase explicit applications of our results, we study the MaxCut problem and conclude that quantum annealing needs at least a run-time that scales logarithmically in the problem size to beat classical algorithms on MaxCut. To establish our results, we also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians which might be of independent interest.