This talk discusses recent and ongoing work on a new "local" perspective on quantitative propagation of chaos, both for exchangeable and non-exchangeable systems. For an exchangeable system of $n$ diffusive particles interacting pairwise, the relative entropy between the marginal law of $k$ particles and its limiting product measure is shown to be $O((k/n)^2)$ at each time, as long as the same is true at time zero, and as long as the interaction kernel is sufficiently regular. Gaussian examples show that this is sharp. In contrast, prior "global" methods are based on the analysis of the full collection of $n$ particles and can yield at best $O(k/n)$. For non-exchangeable systems, more nuanced entropy bounds are obtained in terms of the fine structure of the matrix of pairwise interaction strengths. At the heart of the local approach is a hierarchy of differential inequalities, which, in the exchangeable case, bound the $k$-particle entropy in terms of the $(k+1)$-particle entropy for each $k$. The hierarchy is significantly more complex in the non-exchangeable setting, indexed by sets rather than numbers of particles, and we analyze it by means of an unexpected connection with first-passage percolation.