This proposal aims to determine sharp decay rates for covering numbers and Kolmogorov n-widths on complex spheres, key tools in approximation theory and kernel-based machine learning. The research will address the existence of the asymptotic limit of covering numbers and its implications for the spectral decay of positive integral operators on complex spheres. Two approaches will be explored: one based on the construction of operators via suitable polynomials and another using approximation techniques such as K-functionals and moduli of smoothness. Practical applications illustrating the theoretical results will also be investigated.