  Majorana stars, the $2j$ spin coherent states that are orthogonal to a
spin-$j$ state, offer a visualization of general quantum states and may
disclose deep structures in quantum states and their evolutions. In particular,
the genuine tripartite entanglement - the three-tangle of a symmetric
three-qubit state, which can be mapped to a spin-3/2 state, is measured by the
normalized product of the distance between the Majorana stars. However, the
Majorana representation cannot applied to general non-symmetric $n$-qubit
states. We show that after a series of SL$(2,\mathbb{C})$ transformations,
non-symmetric three-qubit states can be transformed to symmetric three-qubit
states, while at the same time the three-tangle is unchanged. Thus the genuine
tripartite entanglement of general three-qubit states has the geometric
representation of the associated Majorana stars. The symmetrization and hence
the Majorana star representation of certain genuine high-order entanglement for
more qubits are possible for some special states. In general cases, however,
the constraints on the symmetrization may prevent the Majorana star
representation of the genuine entanglement.

Quantum Physics