We compare different notions of curvature on contact sub-Riemannian manifolds.
In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi
equation. The main result is that all these coefficients are encoded in the asymptotic expansion
of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their
expressions in terms of the standard tensors of contact geometry. As an application of these
results, we prove a version of the sub-Riemannian Bonnet-Myers theorem that applies to any
contact manifold, with special attention to contact Yang-Mills structures.