  We show that, for a given compact or discrete quantum group $G$, the class of
actions of $G$ on C*-algebras is first-order axiomatizable in the logic for
metric structures. As an application, we extend the notion of Rokhlin property
for $G$-C*-algebra, introduced by Barlak, Szab\'{o}, and Voigt in the case when
$G$ is second countable and coexact, to an arbitrary compact quantum group $G$.
All the the preservations and rigidity results for Rokhlin actions of second
countable coexact compact quantum groups obtained by Barlak, Szab\'{o}, and
Voigt are shown to hold in this general context. As a further application, we
extend the notion of equivariant order zero dimension for equivariant
*-homomorphisms, introduced in the classical setting by the first and third
authors, to actions of compact quantum groups. This allows us to define the
Rokhlin dimension of an action of a compact quantum group on a C*-algebra,
recovering the Rokhlin property as Rokhlin dimension zero. We conclude by
establishing a preservation result for finite nuclear dimension and finite
decomposition rank when passing to fixed point algebras and crossed products by
compact quantum group actions with finite Rokhlin dimension.
