Interactions in many-body systems are typically short-range and few-body. We investigate how such local interactions build up long-range and intrinsically multipartite entanglement by studying the $n$-partite Greenberger-Horne-Zeilinger $\left({\text{GHZ}}_n\right)$ entanglement in monitored random Clifford circuits, which is well-known for a measurement-induced transition between phases of volume-law and area-law (bipartite) entanglement. We obtain a series of results: (1) About 1.25 $|{\text{GHZ}}_3\rangle$ can be extracted from states in the volume-law phase. This value is remarkably universal, independent of both the measurement rate and partitioning details, until a phase transition (either measurement-induced or a newly identified partitioning-induced transition) is approached. (2) Dynamically, the creation (sometimes also the annihilation) of ${\text{GHZ}}_3$ entanglement occurs suddenly via dynamical phase transitions (DPTs). The critical points of these DPTs are governed by the entanglement speed $\left(v_E\right)$ of biaprtite entanglement. (3) In stark contrast to ${\text{GHZ}}_{n\le 3}$, ${\text{GHZ}}_{n\ge 4}$ entanglement is statistically significant only at the measurement-induced critical point, not in the bulk of the volume-law phase. Our results uncover a rich hierarchy of multipartite entanglement structures.

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• Nonequilibrium statistical mechanics
• Phase transitions
• Quantum circuits
• Quantum entanglement
• Quantum teleportation