  This paper provides the quantum treatment of the relational quadrilateral.
The underlying reduced configuration spaces are $\mathbb{CP}^2$ and the cone
over this, C($\mathbb{CP}^2$). We consider exact free and isotropic HO
potential cases and perturbations about these. Moreover, our purely relational
kinematical quantization is distinct from the usual one for $\mathbb{CP}^2$,
which turns out to carry absolutist connotations instead. Thus this paper is
the first to note absolute-versus-relational motion distinctions at the
kinematical rather than dynamical level, and also an example of value to the
discussion of kinematical quantization along the lines of Isham 1984. This
treatment of the relational quadrilateral is the first relational QM with very
new mathematics for a finite QM model, and is far more typical of the general
quantum relational $N$-a-gon than the previously-studied case of the relational
triangle. We consider useful integrals as regards perturbation theory and the
peaking interpretation of quantum cosmology. We subsequently consider problem
of time applications of this: quantum Kucha\v{r} beables, the Machian version
of the semiclassical approach and the timeless naive Schrodinger
interpretation. These go toward extending the combined Machian
semiclassical-histories-timeless approach of [1] to the case of the
quadrilateral, which will be treated in subsequent papers.
