Geometry on a sphere reveals hidden structure in quantum entanglement

The Majorana stellar representation maps any symmetric multi-qubit or higher-spin state onto a set of points on the Bloch sphere, turning abstract entanglement into concrete geometry. This review consolidates scattered results showing that constellation shapes directly encode entanglement measures — concurrence, three-tangle, genuine multipartite entanglement — and that their evolution reveals anomalous contributions to geometric (Berry/Hannay) phases. A key practical claim is that this geometric approach sidesteps #P-hard complexity, allowing multipartite invariants to be evaluated in polynomial time. Applications span quantum metrology, state engineering, and condensed-matter physics.