  We investigate the evolution of the chiral magnetic instability in a
protoneutron star and compute the resulting magnetic power and helicity
spectra. The instability may act during the early cooling phase of the hot
protoneutron star after supernova core collapse, where it can contribute to the
buildup of magnetic fields of strength up to the order of $10^{14}$ G. The
maximal field strengths generated by this instability, however, depend
considerably on the temperature of the protoneutron star, on density
fluctuations and turbulence spectrum of the medium. At the end of the hot
cooling phase the magnetic field tends to be concentrated around the
submillimeter to cm scale, where it is subject to slow resistive damping.
