We show that uncertainties in the strength of interactions of hyperons among themselves and with nucleons lead to a large uncertainty in the maximum allowed neutron star mass, even if the properties of nuclear and neutron matter are known with infinite precision around normal nuclear matter density and below. The presence of hyperons in the neutron star will generate a φ-meson condensate, however, and this reduces the sensitivity to the strengths of the couplings. The possibility that nucleons have a high strangeness content is explored, but it turns out to have negligible influence on neutron star structure. We consider a novel mechanism for nuclear attraction, a density-dependent glueball condensate. Finally, we determine which of these nuclear equations of state lead to a stable quark matter core in the star, via a first- or second-order phase transition.