Prismatic and $q$-crystalline sites of higher level

In this article, we define the $m$-prismatic site and the $m$-$q$-crystalline site, which are higher-level analogs of the prismatic site and the $q$-crystalline site respectively. We prove a certain equivalence between the category of crystals on the $m$-prismatic site (resp. the $m$-$q$-crystalline site) and that on the prismatic site (resp. the $q$-crystalline site), which can be regarded as the prismatic (resp. the $q$-crystalline) analog of the Frobenius descent due to Berthelot and the Cartier transform due to Ogus–Vologodsky, Oyama and Xu. We also prove equivalence between the category of crystals on the $m$-prismatic site and that on the $(m-1)$-$q$-crystalline site.