The Gaussian Symplectic Ensemble (GSE) is a class of random matrices characterized by quaternionic entries, which are invariant under symplectic transformations, making them relevant in the study of quantum systems with time-reversal symmetry and odd-spin particles. GSE is one of the three classical ensembles in random matrix theory, alongside the Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble, each describing different symmetry classes in quantum physics.