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Quantification of coherence lies at the heart of quantum information processing and fundamental physics. Exact evaluation of coherence measures generally needs a full reconstruction of the density matrix, which becomes intractable for large-scale multipartite systems. Here, we propose a systematic theoretical approach to efficiently estimating lower and upper bounds of coherence in multipartite states. Under the stabilizer formalism, the lower bound is determined by the spectrum estimation method with a small number of measurements, and the upper bound is determined by a single measurement. We verify our theory with a four-qubit optical quantum system. We experimentally implement various multiqubit entangled states, including the Greenberger-Horne-Zeilinger state, the cluster state, and the W state, and show how their coherences are efficiently inferred from measuring a few observables.