Tunable Families of Multiqubit Elegant Joint Measurements

We give a closed-form construction of the n-qubit Elegant Joint Measurement (EJM) proposed in \[PRL 136, 190201 (2026)\] and show that it is part of a tunable family of measurements with tetrahedrally arranged Bloch vectors. The construction is based on the interference pattern implied by a single phase polynomial built from the elementary symmetric functions. It realises a regular tetrahedral measurement for every n, and the corresponding measurement unitary lies at level n{+}1 of the Clifford hierarchy. Starting from this measurement, we ask whether the size of the local tetrahedron - and hence the entanglement of the basis - can be varied while preserving its symmetry. For every even n the answer is yes, and remarkably the size follows the same one-parameter law that governs the known two-qubit family, interpolating down to a 1-uniform basis. For n=3 the EJM is locally isolated, while for odd nge5 we do not know an analogous closed-form family. We also give an analogous construction, valid for every n geq3, with square local geometry.