  This paper is concerned with quantum cohomology and Fukaya categories of a
closed monotone symplectic manifold $X$, where we use coefficients in a field
$\mathbf{k}$ of characteristic $p>0$. The first main result of this paper is
that the quantum Steenrod operations $Q\Sigma$ admit an interpretation in terms
of the Fukaya category of $X$, via suitable versions of the open-closed maps.
Using this, we show that $Q\Sigma$, whose definition is intrinsic to
characteristic $p$, is compatible with certain structures inherited from the
quantum connection in characteristic $0$. We then turn to applications of these
results. The first application is an arithmetic proof of the unramified
exponential type conjecture for $X$ that satisfies Abouzaid's generation
criterion over $\overline{\mathbb{Q}}$, which uses a reduction mod $p$
argument. Next, we demonstrate how the categorical perspective provides new
tools for computing $Q\Sigma$ beyond the reach of known technology. We also
explore potential connections of our work to arithmetic homological mirror
symmetry.
