  We apply the shifted composition rule -- an information-theoretic principle
introduced in our earlier work [AC23] -- to establish shift Harnack
inequalities for the Langevin diffusion. We obtain sharp constants for these
inequalities for the first time, allowing us to investigate their relationship
with other properties of the diffusion. Namely, we show that they are
equivalent to a sharp "local gradient-entropy" bound, and that they imply
curvature upper bounds in a compelling reflection of the Bakry-Emery theory of
curvature lower bounds. Finally, we show that the local gradient-entropy
inequality implies optimal concentration of the score, a.k.a. the logarithmic
gradient of the density.
