Abstract
We construct optimal protocols for verifying qubit and qudit GHZ states using local projective measurements. When the local dimension is a prime, an optimal protocol is constructed from Pauli measurements only. Our protocols provide a highly efficient way for estimating the fidelity and certifying genuine multipartite entanglement. In particular, they enable the certification of genuine multipartite entanglement using only one test when the local dimension is sufficiently large. By virtue of adaptive local projective measurements, we then construct protocols for verifying GHZ-like states that are optimal over all protocols based on one-way communication. The efficiency can be improved further if additional communications are allowed. Finally, we construct optimal protocols for verifying GHZ states and nearly optimal protocols for GHZ-like states in the adversarial scenario.