This paper reports a summary of the results of the PhD thesis. It is divided into two parts. In the first part the dynamics of the simplest ‘atom’ of space, emergent semiclassically from a purely quantum object called the volume operator, is studied. The associated three-terms recurrence relation and the family of orthogonal polynomials are described. In the second part, we study Lorentzian structures needed to build a discretized model of spacetime focusing on the topology $S^3 \times R$.