  Grothendieck's bound is used in the context of a single quantum system, in
contrast to previous work which used it for multipartite entangled systems and
the violation of Bell-like inequalities. Roughly speaking the Grothendieck
theorem considers a `classical' quadratic form ${\cal C}$ that uses complex
numbers in the unit disc, and takes values less than $1$. It then proves that
if the complex numbers are replaced with vectors in the unit ball of the
Hilbert space, then the `quantum' quadratic form ${\cal Q}$ might take values
greater than $1$, up to the complex Grothendieck constant $k_G$. The
Grothendieck theorem is reformulated here in terms of arbitrary matrices (which
are multiplied with appropriate normalisation prefactors), so that it is
directly applicable to quantum quantities. The emphasis in the paper is in the
`Grothendieck region' $(1,k_G)$, which is a classically forbidden region in the
sense that ${\cal C}$ cannot take values in it. Necessary (but not sufficient)
conditions for ${\cal Q}$ taking values in the Grothendieck region are given.
Two examples that involve physical quantities in systems with $6$ and
$12$-dimensional Hilbert space, are shown to lead to ${\cal Q}$ in the
Grothendieck region $(1,k_G)$. They involve projectors of the overlaps of novel
generalised coherent states that resolve the identity and have a discrete
isotropy.
