Is it possible that quasars are not powered by black holes but some other kind of compact, massive object? AFAIK, objects in the universe thought to be black holes show evidence of radiatively inefficient accretion flow. But is this the case with Quasars? If not, is there a possibility that the core of a quasar is an object similar to a neutron star.
The solution to your question is surprisingly simple, I think:

A quasar that puts out energy around Eddington luminosity or higher, must accrete at a certain rate, corresponding to the energy output. 

When consulting a lecture and a random paper from the archive on this topic, it is evident that those accretion rates correspond to 1-10 solar masses per year or higher. A neutron star accreting 1-10 solar masses in one year will become a very massive black hole quickly enough.