Multipartite non-Gaussian entanglement serves as a key physical resource for realizing universal continuous-variable quantum computation. Recently, three-mode spontaneous parametric down-conversion (SPDC), which enables the deterministic generation of non-Gaussian Greenberger-Horne-Zeilinger (GHZ) states, was observed in superconducting circuits, and the non-Gaussian entanglement properties of these states were characterized. However, to date, no criterion capable of detecting genuine $N$-partite non-Gaussian entanglement has been proposed. In this work, we put forward a set of necessary and sufficient conditions for the separability of $N$-mode non-Gaussian GHZ states generated by $N$-mode SPDC. Building on these separability conditions, we further derive a set of genuine $N$-partite non-Gaussian entanglement criteria. Through numerical modeling, we demonstrate that quantum states generated by four-mode SPDC exhibit full inseparability and genuine quadripartite non-Gaussian entanglement. Our results hold significant applications for the characterization and certification of multipartite non-Gaussian entanglement.

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• Entanglement detection
• Photon pairs & parametric down-conversion
• Quantum entanglement