  In this paper, we study the multidimensional lattice Schr\"odinger operators
with $C^2$-cosine like quasi-periodic (QP) potential. We establish quantitative
Green's function estimates, the arithmetic version of Anderson (and dynamical)
localization, and the finite volume version of $(\frac 12-)$-H\"older
continuity of the integrated density of states (IDS) for such QP Schr\"odinger
operators. Our proof is based on an extension of the fundamental multi-scale
analysis (MSA) type method of Fr\"ohlich-Spencer-Wittwer [\textit{Comm. Math.
Phys.} 132 (1990): 5--25] to the higher lattice dimensions. We resolve the
level crossing issue on eigenvalues parameterizations in the case of both
higher lattice dimension and $C^2$ regular potential.
