A stacky approach to prismatic crystals via $q$-prism charts

Let $Y$ be a locally complete intersection over $\mathcal{O}_K$ containing a $p$-power root of unity $\zeta_p$. We classify the derived category of prismatic crystals on the absolute prismatic site of $Y$ by studying quasi-coherent complexes on the prismatization of $Y$ via $q$-prism charts. We also develop a Galois descent mechanism to remove the assumption on $\mathcal{O}_K$. As an application, we classify quasi-coherent complexes on the Cartier-Witt stack and give a purely algebraic calculation of the cohomology of the structure sheaf on the absolute prismatic site of $\mathbb{Z}_p$. Along the way, for $Y$ a locally complete intersection over $\overline{A}$ with $A$ lying over a $q$-prism, we classify quasi-coherent complexes on the relative prismatization of $Y$.