Multipartite high-dimensional entanglement improves quantum teleportation with information sharing, controlled reconstruction, higher capacity, and stronger noise resilience. To ensure the validity and strength of these capabilities, it is essential to certify whether an $N$-partite high-dimensional teleportation process employs genuine $N$-partite entanglement of at least a certain dimension. Here we propose a robustness-based criterion that can reliably identify the minimum entanglement dimension necessary to reproduce the observed teleportation data in a teleportation process, outperforming traditional fidelity-based measures that cannot distinguish genuine multipartite entanglement from biseparable simulations. We numerically demonstrate the criterion's effectiveness in noisy teleportation scenarios and its utility for securely evaluating controller authority in controlled quantum teleportation. This work establishes a practical benchmark for verifying quantum advantages in multipartite high-dimensional quantum networks.

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• Quantum information theory
• Quantum networks
• Quantum teleportation
• Quantum verification