Nonequilibrium phenomena occupy a prominent role at the frontiers of physics. Since its conception in the mid-70s, the Kibble-Zurek mechanism has been the paradigmatic framework to describe the dynamics of phase transition, in which symmetry breaking leads to the formation of topological defects (e.g. vortices in a superfluid or kinks in a spin chain). Its key testable prediction is that the average number of topological defects scales as a universal power law with the quench rate (ie. the velocity at which the critical point is crossed). We unveil signatures of universality beyond the mean number of topological defects and show that the full counting statistics of topological defects is actually unanimous.

We thoroughly investigate the fingerprint of equilibrium quantum phase transitions through the single-site two-time correlations and violation of Leggett-Garg inequalities in spins-1/2 and Majorana Fermion systems. By means of simple analytical arguments for a general spin-1/2 Hamiltonian, and matrix product simulations of one-dimensional XXZ and anisotropic XY models, we argue that finite-order quantum phase transitions can be determined by singularities of the time correlations or their derivatives at criticality. The same features are exhibited by corresponding Leggett-Garg functions, which noticeably indicate violation of the Leggett-Garg inequalities for early times and all the Hamiltonian parameters considered.