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The geometric measure of entanglement is investigated for permutationsymmetric pure states of multipartite qubit systems, in particular the questionof maximum entanglement. This is done with the help of the Majoranarepresentation, which maps an n qubit symmetric state to n points on the unitsphere. It is shown how symmetries of the point distribution can be exploitedto simplify the calculation of entanglement and also help find the maximallyentangled symmetric state. Using a combination of analytical and numericalresults, the most entangled symmetric states for up to 12 qubits are exploredand discussed. The optimization problem on the sphere presented here is thencompared with two classical optimization problems on the S^2 sphere, namelyToth's problem and Thomson's problem, and it is observed that, in general, theyare different problems.

The maximally entangled symmetric state in terms of the geometric measure