  We prove the optimal rate of quantitative propagation of chaos, uniformly in
time, for interacting diffusions. Our main examples are interactions governed
by convex potentials and models on the torus with small interactions. We show
that the distance between the $k$-particle marginal of the $n$-particle system
and its limiting product measure is $O((k/n)^2)$, uniformly in time, with
distance measured either by relative entropy, squared quadratic Wasserstein
metric, or squared total variation. Our proof is based on an analysis of
relative entropy through the BBGKY hierarchy, adapting prior work of the first
author to the time-uniform case by means of log-Sobolev inequalities.
