  We propose that the recent observations reported by the different Pulsar
Timing Array (PTA) collaborations (i.e.~IPTA, EPTA, PPTA, and NANOGrav) of a
common process over several pulsars could correspond to a stochastic
gravitational wave background (SGWB) produced by turbulent sources in the early
universe, in particular due to the magnetohydrodynamic (MHD) turbulence induced
by primordial magnetic fields. I discuss recent results of numerical
simulations of MHD turbulence and present an analytical template of the SGWB
validated by the simulations. We use this template to constrain the magnetic
field parameters using the results reported by the PTA collaborations. Finally,
we compare the constraints on the primordial magnetic fields obtained from PTA
with those from blazar signals observed by Fermi Large Area Telescope (LAT),
from ultra high-energy cosmic rays, and from the cosmic microwave background.
We show that a non-helical primordial magnetic field produced at the scale of
the quantum chromodynamics phase transition is compatible with such constraints
and it could additionally provide with a magnetic field at recombination that
would help to alleviate the Hubble tension.
