The nuclear equation of state is an important component in the evolution of core collapse supernovae. In this paper we make a survey of various equations of state in the literature and analyze their effect on spherical core-collapse models in which the effects of three-dimensional turbulence is modeled by a general relativistic formulation of Supernova Turbulence in Reduced dimensionality (STIR). We show that the viability of the explosion is quite EOS dependent and that it best correlates with the early-time interior entropy density of the proto-neutron star. We check that this result is not progenitor dependent, although low-mass progenitors show different explosion properties, due to the different pre-collapse nuclear composition. Larger central entropies also induce more vigorous proto-neutron-star convection in our one-dimensional turbulence model, as well as a wider convective layer.