  Flatness -- the absence of spacetime curvature -- is a well-understood
property of macroscopic, classical spacetimes in general relativity. The same
cannot be said about the concepts of curvature and flatness in nonperturbative
quantum gravity, where the microscopic structure of spacetime is not
describable in terms of small fluctuations around a fixed background geometry.
An interesting case are two-dimensional models of quantum gravity, which lack a
classical limit and therefore are maximally "quantum". We investigate the
recently introduced quantum Ricci curvature in CDT quantum gravity on a
two-dimensional torus, whose quantum geometry could be expected to behave like
a flat space on suitably coarse-grained scales. On the basis of Monte Carlo
simulations we have performed, with system sizes of up to 600.000 building
blocks, this does not seem to be the case. Instead, we find a scale-independent
"quantum flatness", without an obvious classical analogue. As part of our
study, we develop a criterion that allows us to distinguish between local and
global, topological properties of the toroidal quantum system.
