  In several approaches towards a quantum theory of gravity, such as group
field theory and loop quantum gravity, quantum states and histories of the
geometric degrees of freedom turn out to be based on discrete spacetime. The
most pressing issue is then how the smooth geometries of general relativity,
expressed in terms of suitable geometric observables, arise from such discrete
quantum geometries in some semiclassical and continuum limit. In this thesis I
tackle the question of suitable observables focusing on the effective dimension
of discrete quantum geometries. For this purpose I give a purely combinatorial
description of the discrete structures which these geometries have support on.
As a side topic, this allows to present an extension of group field theory to
cover the combinatorially larger kinematical state space of loop quantum
gravity. Moreover, I introduce a discrete calculus for fields on such
fundamentally discrete geometries with a particular focus on the Laplacian.
This permits to define the effective-dimension observables for quantum
geometries. Analysing various classes of quantum geometries, I find as a
general result that the spectral dimension is more sensitive to the underlying
combinatorial structure than to the details of the additional geometric data
thereon. Semiclassical states in loop quantum gravity approximate the classical
geometries they are peaking on rather well and there are no indications for
stronger quantum effects. On the other hand, in the context of a more general
model of states which are superposition over a large number of complexes, based
on analytic solutions, there is a flow of the spectral dimension from the
topological dimension $d$ on low energy scales to a real number $0<\alpha<d$ on
high energy scales. In the particular case of $\alpha=1$ these results allow to
understand the quantum geometry as effectively fractal.
