Recent developments in approximate counting have made startling progress indeveloping fast algorithmic methods for approximating the number of solutionsto constraint satisfaction problems (CSPs) with large arities . However, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood . We focus on the problem of approximately counting$q$-colourings on $K$-uniform hypergraphs with bounded degree $Delta$ Anefficient algorithm exists if $\Delta\lesssim \frac{q^{K/3-1,1,4^KK^2$ (Jain,Pham, and Voung, 2021; He, Sun, and Wu, 2021) Ahardness bound is not known even for the easier problem of finding colourings . For all even $q\geq 4$ it is NP-hard to approximate the numberof colourings when $Delta\gtrsim q^{Kq^K/2}$ is $2/2/3/2 . We show inparticular that for all . Even$q/3+\$

Author(s) : Andreas Galanis, Heng Guo, Jiaheng Wang