In this talk, we present certain results regarding the higher genus Gromov-Witten theory of $\mathbb{C}^n/\mathbb{Z}_n$ obtained by studying its cohomological field theory structure in detail. Holomorphic anomaly equations are certain recursive partial differential equations predicted by physicists for the Gromov-Witten potential of a Calabi-Yau threefold. We prove holomorphic anomaly equations for $\mathbb{C}^n/\mathbb{Z}_n$ for any $n\geq 3$. In other words, we present a phenomenon of holomorphic anomaly equations in arbitrary dimension, a result beyond the consideration of physicists. The proof of this fact relies on showing that the Gromov-Witten potential of $\mathbb{C}^n/\mathbb{Z}_n$ lies in a certain polynomial ring. This talk is based on the joint work arXiv:2301.08389 with Hsian-Hua Tseng.