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 is given by the 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 causal dynamical triangulations quantum gravity on a two-dimensional torus, whose quantum geometry could be expected to behave as 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 analog. 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.