Maximal estimates for orthonormal systems in the context of the Schrödinger equation were recently developed by Bez, Lee, and Nakamura as a natural analog to the corresponding classical problem for a single initial datum. In this talk, we will focus on wave equations and investigate maximal estimates for orthonormal systems. While the classical problem for a single datum is relatively straightforward, the orthonormal system version presents significantly more challenges. Nonetheless, we have obtained some partial results through geometric observations. This work is a collaboration with Shinya Kinoshita and Hyerim Ko.