Pseudo-rotations and Steenrod squares

In this note we prove that if a closed monotone symplectic manifold $M$ of dimension $2n,$ satisfying a homological condition, that holds in particular when the minimal Chern number is $N>n,$ admits a Hamiltonian pseudorotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudorotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.