Arithmetic $k$-Polynomial Dynamical Localization for $k$ Power-Law Quasi-Periodic Long-Range Operators on $\ell^2(\mathbb{Z}^d)$

We establish a criterion of arithmetic $k$-polynomial spectral localization and $k$-polynomial dynamical localization in expectation for quasi-periodic long-range operators on $\ell^2(\mathbb{Z}^{d})$ with power-law hopping based on the quantitative $C^{k}$-reducibility of the dual Schrödinger cocyc