Kaledin established a Cartier isomorphism for cyclic homology of dg-categories over fields of characteristic p, generalizing a classical construction in algebraic geometry. In joint work with Paul Seidel, we showed that this isomorphism and related results imply concrete statements about the structure of quantum connections on monotone symplectic manifolds (both in characteristic p and characteristic zero). I will explain these results and, if time permits, I will also describe some open questions concerning the enumerative interpretation of the Cartier isomorphism as well as connections to quantum Steenrod operations.