  We investigate the dependence of the conductivity and the entanglement
entropy on the space-time dimensionality $d$ in two holographic
superconductors: one dual to a quantum critical point with spontaneous symmetry
breaking, and the other modeled by a charged scalar that condenses at a
sufficiently low temperature in the presence of a Maxwell field. In both cases
the gravity background is asymptotically Anti de Sitter (AdS). In the large $d$
limit we obtain explicit analytical results for the conductivity at zero
temperature and the entanglement entropy by a $1/d$ expansion. We show that the
entanglement entropy is always smaller in the broken phase. As dimensionality
increases, the entanglement entropy decreases, the coherence peak in the
conductivity becomes narrower and the ratio between the energy gap and the
critical temperature decreases. These results suggest that the condensate
interactions become weaker in high spatial dimensions.
