Topological entropy is one of the fundamental invariants of a dynamical system, measuring its complexity. In this talk, we discuss connections between the topological entropy of a Hamiltonian dynamical system, e.g., a geodesic flow or a Hamiltonian diffeomorphism, and the Morse/Floer homology filtered by the action and associated to it via a variational framework in a very broad sense. We introduce a new invariant associated with the system, the barcode entropy. We show that barcode entropy is closely related to topological entropy and that these invariants are equal in low dimensions. The talk is based on joint work with Erman Cineli, Basak Gurel and Marco Mazzucchelli.