  We perform a forecast analysis on how well a Euclid-like photometric galaxy
cluster survey will constrain the total neutrino mass and effective number of
neutrino species. We base our analysis on the Monte Carlo Markov Chains
technique by combining information from cluster number counts and cluster power
spectrum. We find that combining cluster data with CMB measurements from Planck
improves by more than an order of magnitude the constraint on neutrino masses
compared to each probe used independently. For the LCDM+m_nu model the 2 sigma
upper limit on total neutrino mass shifts from M_nu < 0.35 eV using cluster
data alone to M_nu < 0.031 eV when combined with CMB data. When a non-standard
model with N_eff number of neutrino species is considered, we estimate
N_eff<3.14 (95% CL), while the bounds on neutrino mass are relaxed to M_nu <
0.040 eV. This accuracy would be sufficient for a 2 sigma detection of neutrino
mass even in the minimal normal hierarchy scenario. We also consider scenarios
with a constant dark energy equation of state and a non-vanishing curvature.
When these models are considered the error on M_nu is only slightly affected,
while there is a larger impact of the order of ~ 15 % and ~ 20% respectively on
the 2 sigma error bar of N_eff with respect to the standard case. We also treat
the LCDM+m_nu+N_eff case with free nuisance parameters, which parameterize the
uncertainties on the cluster mass determination. In this case, the upper bounds
on M_nu are relaxed by a factor larger than two, M_nu < 0.083 eV (95% CL),
hence compromising the possibility of detecting the total neutrino mass with
good significance. We thus confirm the potential that a large optical/near-IR
cluster survey, like that to be carried out by Euclid, could have in
constraining neutrino properties [abridged].
