Constraints on four-party entanglement in holography

We show that in pure time-reflection-symmetric holographic states all known four-party entanglement signals vanish unless the triple information $I_3$ is non-zero. In this sense, our results show that $I_3$ is the strongest known signal of the presence of quadripartite entanglement. Additionally, $I_3$ quantitatively bounds all four-party entanglement signals built from the multi-entropy. However, the residual entropy $Q_4$, also a measure of four-party entanglement, is not bounded by $I_3$, although $I_3=0$ does imply $Q_4=0$ for holographic states (except on a set of measure zero for which $Q_4$ is ill-defined).