In recent decades, quantitative measures of entanglement have become vital tools in characterizing quantum phases of matter. This is especially true for topological phases, like quantum Hall states, which lack local order parameters and are instead best characterized by their entanglement structure. For instance, the entanglement entropy, a standard measure of entanglement, is able to detect topologically ordered phases, which support fractionalized anyon quasiparticles, like fractional quantum Hall states. However, the entanglement entropy is only sensitive to bipartite (i.e. Bell pair-like entanglement). In this talk, I will present our recent results going beyond the entanglement entropy in characterizing the tripartite entanglement structure of topological phases. Remarkably, we find that topological phases with gapless chiral edge modes support a distinct form of tripartite entanglement from those with gapped edges. Our work hints at a rich structure of multipartite correlations in topological phases, which has largely been unexplored.