  Our Universe is ruled by quantum mechanics and should be treated as a quantum
system. {$SU(\infty)$-QGR is a recently proposed quantum model for the
Universe, in which gravity is associated to $SU(\infty)$ symmetry of its
Hilbert space. Clustering/blockization of its infinite dimensional state due to
random quantum fluctuations divides the Universe to approximately isolated
subsystems. In addition to parameters of their {\it internal} finite rank
symmetries, states and dynamics of subsystems are characterized by 4 continuous
parameters, and the perceived classical spacetime is their effective
representation, reflecting quantum states of subsystems and their relative
evolution. At lowest order the effective Lagrangian of {$SU(\infty)$-QGR has
the form of Yang-Mills gauge theories for both $SU(\infty)$ - gravity - and
internal symmetries defined on the aforementioned 4D parameter space. In the
present work we study more thoroughly some of the fundamental aspects of
{$SU(\infty)$-QGR. Specifically, we clarify impact of the degeneracy of
$\mathcal{SU}(\infty)$; describe mixed states of subsystems and their
purification; calculate measures of their entanglement to the rest of the
Universe; and discuss their role in the emergence of local gauge symmetries. We
also describe the relationship between what is called {\it internal space} of
$SU(\infty)$ Yang-Mills with the 4D parameter space, and analytically
demonstrate irrelevance of the geometry of parameter space for physical
observables. Along with these topics, we demonstrate the equivalence of two
sets of criteria for compositeness of a quantum system, and show uniqueness of
the limit of various algebras leading to $\mathcal{SU}(\infty)$.
