  Twenty years ago Cayley's hyperdeterminant, the degree four invariant of the
polynomial ring $\mathbb{C}[\mathbb{C}^2\otimes\mathbb{C}^2\otimes
\mathbb{C}^2]^{{\text{SL}_2(\mathbb{C})}^{\times 3}}$, was popularized in
modern physics as separates genuine entanglement classes in the three qubit
Hilbert space and is connected to entropy formulas for special solutions of
black holes. In this note we compute the analogous invariant on the fermionic
Fock space for $N=8$, i.e. spin particles with four different locations, and
show how this invariant projects to other well-known invariants in quantum
information. We also give combinatorial interpretations of these formulas.
