  This work is devoted to the study of integral $p$-adic Hodge theory in the
context of Artin stacks. For a Hodge-proper stack, using the formalism of
prismatic cohomology, we establish a version of $p$-adic Hodge theory with the
\'etale cohomology of the Raynaud generic fiber as an input. In particular, we
show that the corresponding Galois representation is crystalline and that the
associated Breuil-Kisin module is given by the prismatic cohomology. An
interesting new feature of the stacky setting is that the natural map between
\'etale cohomology of the algebraic and the Raynaud generic fibers is often an
equivalence even outside of the proper case. In particular, we show that this
holds for global quotients $[X/G]$ where $X$ is a smooth proper scheme and $G$
is a reductive group. As applications we deduce Totaro's conjectural inequality
and also set up a theory of $A_{\mathrm{inf}}$-characteristic classes.
