An important object in the study of high-dimensional non-convex functions is the so-called complexity, defined as the logarithm of the number of critical points. For random functions, this is typically obtained by the Kac-Rice formula, which requires in particular the computation of certain random Hessian determinants. In this talk I will present some recent progresses concerning the complexity of the Thouless-Anderson-Palmer (TAP) energy associated to the celebrated Sherrington-Kirkpatrick model for spin glasses. A joint work with D. Belius, F. Concetti.