The origin and evolution of magnetic fields of neutron stars from birth have long been a source of debate. Here, motivated by recent simulations of the Hall cascade with magnetic helicity, we invoke a model where the largescale magnetic field of neutron stars grows as a product of small-scale turbulence through an inverse cascade. We apply this model to a simulated population of neutron stars at birth and show how this model can account for the evolution of such objects across the PP. diagram, explaining both pulsar and magnetar observations. Under the assumption that small-scale turbulence is responsible for large-scale magnetic fields, we place a lower limit on the spherical harmonic degree of the energy-carrying magnetic eddies of approximate to 40. Our results favor the presence of a highly resistive pasta layer at the base of the neutron star crust. We further discuss the implications of this paradigm on direct observables, such as the nominal age and braking index of pulsars.