Quantum correlations, often observed as violations of Bell inequalities, are critical to our understanding of the quantum world, with far-reaching technological and fundamental impact. Many tests of Bell inequalities have studied pairs of correlated particles. However, interest in multi-particle quantum correlations is driving the experimental frontier to test larger systems. All violations to date require supplementary assumptions that open results to loopholes, the closing of which is one of the most important challenges in quantum science. Seminal experiments have closed some loopholes, but no experiment has closed locality loopholes with three or more particles. Here, we close both the locality and freedom-of-choice loopholes by distributing three-photon Greenberger–Horne–Zeilinger entangled states to independent observers. We measured a violation of Mermin's inequality18 with parameter 2.77 ± 0.08, violating its classical bound by nine standard deviations. These results are a milestone in multi-party quantum communication and a significant advancement of the foundations of quantum mechanics.