  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 \emph{concentrated} and
satisfy an \emph{isoperimetric inequality}. To showcase explicit applications
of our results, we study the \textsc{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 \textsc{MaxCut}. To establish our results,
we also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians
which might be of independent interest.
