  For a prime $p>2$ and a smooth proper $p$-adic formal scheme $X$ over
$\mathcal{O}_K$ where $K$ is a $p$-adic field, we study a series of conditions
($\mathrm{Cr}_s$), $s\geq 0$ that partially control the $G_K$-action on the
image of the associated Breuil-Kisin prismatic cohomology
$\mathrm{R}\Gamma_{\Delta}(X/\mathfrak{S})$ inside the
$A_{\mathrm{inf}}$-prismatic cohomology
$\mathrm{R}\Gamma_{\Delta}(X_{A_{\mathrm{inf}}}/A_{\mathrm{inf}})$. The
condition ($\mathrm{Cr}_0$) is a criterion for a Breuil-Kisin-Fargues
$G_K$-module to induce a crystalline representation used by Gee and Liu, and
thus leads to a proof of crystallinity of
$\mathrm{H}^i_{\text{\'{e}t}}(X_{\overline{\eta}}, \mathbb{Q}_p)$ that avoids
the crystalline comparison. The higher conditions ($\mathrm{Cr}_s$) are used to
adapt the strategy of Caruso and Liu in order to establish ramification bounds
for the mod $p$ representations
$\mathrm{H}^{i}_{\text{\'{e}t}}(X_{\overline{\eta}}, \mathbb{Z}/p\mathbb{Z}),$
for arbitrary $e$ and $i$, which extend or improve existing bounds in various
situations.
