  Massive Black Hole (MBH) seeds at redshift $z \gtrsim 10$ are now thought to
be key ingredients to explain the presence of the super-massive ($10^{9-10} \,
\mathrm{M_{\odot}}$) black holes in place $ < 1 \, \mathrm{Gyr}$ after the Big
Bang. Once formed, massive seeds grow and emit copious amounts of radiation by
accreting the left-over halo gas; their spectrum can then provide crucial
information on their evolution. By combining radiation-hydrodynamic and
spectral synthesis codes, we simulate the time-evolving spectrum emerging from
the host halo of a MBH seed with initial mass $10^5 \, \mathrm{M_{\odot}}$,
assuming both standard Eddington-limited accretion, or slim accretion disks,
appropriate for super-Eddington flows. The emission occurs predominantly in the
observed infrared-submm ($1-1000 \, \mathrm{\mu m}$) and X-ray ($0.1 - 100 \,
\mathrm{keV}$) bands. Such signal should be easily detectable by JWST around
$\sim 1 \, \mathrm{\mu m}$ up to $z \sim 25$, and by ATHENA (between $0.1$ and
$10 \, \mathrm{keV}$, up to $z \sim 15$). Ultra-deep X-ray surveys like the
Chandra Deep Field South could have already detected these systems up to $z
\sim 15$. Based on this, we provide an upper limit for the $z \gtrsim 6$ MBH
mass density of $\rho_{\bullet} \lesssim 2.5 \times 10^{2} \, \mathrm{M_{\odot}
\, Mpc^{-3}}$ assuming standard Eddington-limited accretion. If accretion
occurs in the slim disk mode the limits are much weaker, $\rho_{\bullet}
\lesssim 7.6 \times 10^{3} \, \mathrm{M_{\odot} \, Mpc^{-3}}$ in the most
constraining case.
