  We construct a stable homotopy refinement of quantum annular homology, a link
homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$
we associate to an annular link $L$ a naive
$\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to
the quantum annular homology of $L$ as modules over
$\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant
version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The
quotient under the cyclic group action is shown to recover the stable homotopy
refinement of annular Khovanov homology. We study spectrum level lifts of
structural properties of quantum annular homology.
