The existence of magnetic fields is ubiquitous on astrophysical and cosmological scales: from planets andstars to galaxies and galaxy clusters. It is commonly assumed that the observed fields today are originated from either astrophysical or cosmological magnetic seeds.The recent observations of blazar spectra by the Fermi Gamma Ray Observatory provides an intriguing possibility of detecting very weak magnetic fields in cosmic voids.This poses an exciting avenue for studying the generation mechanisms and evolution of observed large-scale correlated magnetic fields. Notably, numerical (cosmological) simulations and faraday rotation measure maps show the large scale morphology of these fields, i.e the magnetic correlation lengths extending beyond the galaxy clusters’ scales. This, in principle, is hard to explain by the astrophysical sources of the field generation and amplification such as the Biermann battery and dynamo even with the various mechanisms of magnetic seed transport (in a few Gyr timeframe). On the other hand, primordial magnetic fields (PMFs), being good candidates for the seed magnetic fields, might be generated in the early Universe through different processes such as quantum-mechanical fluctuations during inflation, bubble collisions during cosmological first order phase transitions, primordial turbulent motions, etc. Interestingly, inflationary generated magnetic fields might have unlimited (i.e.not constrained by the Hubble scale) correlations length scale, while causally generated magnetic fields (for example during the phase transitions) are characterized by the correlation length having an upper limit equal to the Hubble length scale. In our work, using numerical magnetized cosmological simulations we explore the evolution of the primordial magnetic fields (assuming various models of the field generation) during the structure formation (i.e. late stages). We properly account for the magnetic field dynamics prior recombination as well as development of turbulent motions. We study how these seed magnetic fields evolve during structure formation and what can be the observable traces of such fields. Our findings include: the distinctive spectral evolution of different seed fields imprinted on the scale of galaxy clusters, bridges, as well as filaments and voids, and differences in the rotation measure maps.