  The truncation in the number of single-trace chiral primary operators of
$\N=4$ SYM and its conjectured connection with gravity on quantum spacetimes
are elaborated. The model of quantum spacetime we use is $AdS^5_q \times S^5_q$
for $q$ a root of unity. The quantum sphere is defined as a homogeneous space
with manifest $SU_q(3)$ symmetry, but as anticipated from the field theory
correspondence, we show that there is a hidden $SO_q(6)$ symmetry in the
constrution. We also study some properties of quantum space quotients as
candidate models for the quantum spacetime relevant for some $Z_n$ quiver
quotients of the $\N=4$ theory which break SUSY to $\N=2$. We find various
qualitative agreements between the proposed models and the properties of the
corresponding finite $N$ gauge theories.
