  How complex is the structure of quantum geometry? In several approaches, the
spacetime atoms are obtained by the SU(2) intertwiner called quantum
tetrahedron. The complexity of this construction has a concrete consequence in
recent efforts to simulate such models and toward experimental demonstrations
of quantum gravity effects. There are, therefore, both a computational and an
experimental complexity inherent to this class of models. In this paper, we
study this complexity under the lens of stabilizer entropy (SE). We calculate
the SE of the gauge-invariant basis states and its average in the SU(2) gauge
invariant subspace. We find that the states of definite volume are singled out
by the (near) maximal SE and give precise bounds to the verification protocols
for experimental demonstrations on available quantum computers.
