  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 multi-qubit entangled states, including the
Greenberger-Horne-Zeilinger state, the cluster state, and the W state, and show
how their coherence are efficiently inferred from measuring few observables.
