  {\bf Abstract.} We show that two hierarchies of spin Hamilton operators admit
the same spectrum. Both Hamilton operators play a central role for quantum
gates in particular for the case spin-$\frac12$ and the case spin-1. The
spin-$\frac12$, spin-1, spin-$\frac32$ and spin-2 cases are studied in detail.
Entanglement and mutually unbiased bases of the eigenvectors is discussed. Two
triple Hamilton operators are also investigated. Both are also admitting the
same spectrum.
