Topological entropy is one of the fundamental invariants of a dynamical system, measuring the orbit complexity. In this talk, we discuss a connection between the topological entropy of compactly supported Hamiltonian diffeomorphisms and Floer theory. We introduce a new invariant associated with the Floer complexes of the iterates of such a diffeomorphism, which we call barcode entropy. We show that barcode entropy is closely related to topological entropy and that these invariants are equal in dimension two. The talk is based on joint work with Erman Cineli and Viktor Ginzburg.