  We construct a quantum deformation of the Steenrod square construction on
closed monotone symplectic manifolds, based on the work of Fukaya, Betz and
Cohen. We prove quantum versions of the Cartan and Adem relations. We compute
the quantum Steenrod squares for all CP n and give the means of computation for
all toric varieties. As an application, we also describe two examples of
blowups along a subvariety, in which a quantum correction of the Steenrod
square on the blowup is determined by the classical Steenrod square on the
subvariety.
