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The origin of interstellar and intergalactic magnetic fields remains largelyunknown. One possibility is that they are related to the primordial magneticfields (PMFs) produced by, for instance, the phase transitions of the earlyUniverse. In this letter, we show that the PMF-induced turbulence generated ataround the QCD phase transition epoch -- the characteristic magnetic fieldstrength $B_{\rm ch}^* \sim \mathcal{O}(1)~\rm{\mu G}$ and coherent lengthscale $\ell_{\rm ch}^* \sim \mathcal{O}(1)~\rm{pc}$, can naturally accommodatewith nanohertz gravitational waves reported by pulsar timing arraycollaborations. Moreover, the evolution of the PMFs to the recombination erawith the form of $B_{\rm ch}\sim \ell_{\rm ch}^{-\alpha}$ can induce baryondensity inhomogeneities, alter the recombination history, and alleviate thetension of the Hubble parameter $H_0$ and the matter clumpiness parameter $S_8$between early and late-time measurements for $0.88\leq \alpha \leq 1.17$(approximate 95\% credible region based on three PTA likelihoods). The furtherevolved PMFs may account for the $\sim {\cal O}(10^{-16})$ Gauss extra-galacticmagnetic field inferred with GRB 221009A.

Primordial magnetic field as a common solution of nanohertz gravitational waves and Hubble tension