  In the context of symplectic dynamics, pseudo-rotations are Hamiltonian
diffeomorphisms with finite and minimal possible number of periodic orbits.
These maps are of interest in both dynamics and symplectic topology. We show
that a closed, monotone symplectic manifold, which admits a non-degenerate
pseudo-rotation, must have a deformed quantum Steenrod square of the top degree
element, and hence non-trivial holomorphic spheres. This result (partially)
generalizes a recent work by Shelukhin and complements the results by the
authors on non-vanishing Gromov-Witten invariants of manifolds admitting
pseudo-rotations.
