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Permutation-symmetric quantum states appear in a variety of physicalsituations, and they have been proposed for quantum information tasks. Thisarticle builds upon the results of [New J. Phys. 12, 073025 (2010)], where themaximally entangled symmetric states of up to twelve qubits were explored, andtheir amount of geometric entanglement determined by numeric and analyticmeans. For this the Majorana representation, a generalization of the Blochsphere representation, can be employed to represent symmetric n qubit states byn points on the surface of a unit sphere. Symmetries of this point distributionsimplify the determination of the entanglement, and enable the study of quantumstates in novel ways. Here it is shown that the duality relationship ofPlatonic solids has a counterpart in the Majorana representation, and that ingeneral maximally entangled symmetric states neither correspond to anticoherentspin states nor to spherical designs. The usability of symmetric states asresources for measurement-based quantum computing is also discussed.

Geometric Entanglement of Symmetric States and the Majorana Representation