I will explain a formula expressing a compatibility between E_2 algebra structure on Topological Hochschild Cohomology and the cyclotomic structure on Topological Hochschild Homology. A relative analog of this formula specializes to a formula articulated by Bezrukavnikov and Kaledin in their work on quantization in positive characteristic, which expresses the conjugation of the interior product operator by the Cartier isomorphism in terms of Cartan calculus. The inspiration for the noncommutative result comes from my earlier work in equivariant symplectic geometry, which gives an elementary ‘picture-proof’ of the corresponding formula for symplectic invariants. By specializing this result in noncommutative geometry to certain Fukaya categories, e.g. to the Fukaya category of the quintic threefold, one can show that the p-curvature of the mod-p reduction of the quantum connection has an enumerative interpretation: it is a generating series of Z/pZ domain-equivariant Gromov–Witten invariants called the ‘quantum Steenrod operation’, which q-deforms the usual Steenrod operation on cohomology. This gives the first computations of such operations in any Calabi–Yau symplectic manifold admitting a nonconstant holomorphic sphere. The method suggests the existence of an ‘arithmetic realization’ of quantum cohomology by enhancing it with algebraically meaningful Frobenius operators, with rich connections to arithmetic aspects of geometric representation theory.