String theory indicates the existence of primordial Kerr superspinars, extremely compact objects with exterior described by the Kerr naked-singularity geometry. The primordial superspinars have to be converted to a black hole due to accretion, but they could survive to the era of high-redshift quasars. We discuss observational phenomena caused by the primordial Kerr superspinars in this era, considering the properties of corotating Keplerian accretion discs orbiting such superspinars and the optical phenomena modified by their presence. The potential well around a near-extreme superspinar with spin a very close to the extreme black hole value a = 1 is very deep so that the efficiency of the accretion process reaches 157.7%, influencing thus significantly the spectral continuum of corotating Keplerian discs and giving a signature of near-extreme superspinars. Such superspinars can also serve as an efficient accelerator for extremely high-energy collisions. Phenomena enabling a clear distinction of primordial Kerr superspinars and black holes are related to the disc oscillations with the radial and vertical epicyclic frequencies and the most profound could be differences implied by the profiled spectral lines generated in the innermost parts of the corotating Keplerian discs.