Let A and B be local operators in Hamiltonian quantum systems with N degrees of freedom and finite-dimensional Hilbert space. We prove that the commutator norm ‖ A ( t ) , B ‖ is upper bounded by a topological combinatorial problem: counting irreducible weighted paths between two points on the Hamiltonian’s factor graph. Our bounds sharpen existing Lieb–Robinson bounds by removing extraneous growth. In quantum systems drawn from zero-mean random ensembles with few-body interactions, we prove stronger bounds on the ensemble-averaged out-of-time-ordered correlator E ‖ A ( t ) , B ‖ F 2 . In such quantum systems on Erdös–Rényi factor graphs, we prove that the scrambling time t s , at which ‖ A ( t ) , B ‖ F = Θ ( 1 ) , is almost surely t s = Ω ( log N ) ; we further prove t s = Ω ( log N ) to high order in perturbation theory in 1/ N . We constrain infinite temperature quantum chaos in the q -local Sachdev-Ye-Kitaev model at any order in 1/ N ; at leading order, our upper bound on the Lyapunov exponent is within a factor of 2 of the known result at any q > 2 . We also speculate on the implications of our theorems for conjectured holographic descriptions of quantum gravity.