  $SU(\infty)$-QGR is a foundationally quantum approach to gravity. It assumes
that the Hilbert space of the Universe as a whole represents the symmetry group
$SU(\infty)$, and demonstrates this symmetry for Hilbert spaces of infinite
number of subsystems that randomly emerge and represent arbitrary finite rank
internal symmetries. It is shown that their states depend on a dimensionful
parameter arising due to the breaking of a global $U(1)$ symmetry, and a
relative dynamics can be defined by selection of a subsystem as clock. The
action has the form of a Yang-Mills quantum field theory defined on the
(3+1)-dimensional parameter space for both $SU(\infty)$ - gravity - and
internal symmetries. Therefore, $SU(\infty)$-QGR is renormalizable, but
predicts a spin-1 mediator for quantum gravity. Nonetheless, it is proved that
when quantum gravity effects are not detectable, dynamics is similar to the
Einstein-Hilbert action. The aim of present work is in depth study of the
foundation and properties of this model. In particular, we show that the global
$SU(\infty)$ symmetry manifests itself through the entanglement of subsystems
with the rest of the Universe, and demonstrate irrelevance of geometry of the
parameter space for observables. Hence, $SU(\infty)$-QGR deviates from
gauge-gravity duality models, because the classical spacetime is an emergent
effective geometry with negative signature, reflecting relative variation of
quantum states of subsystems and quantum uncertainties. We also briefly discuss
$SU(\infty)$-QGR specific models for dark energy.
