  The assembly of massive black holes in the early universe remains a poorly
constrained open question in astrophysics. The merger and accretion of light
seeds (remnants of Population III stars with mass below $\sim 1000M_{\odot}$)
or heavy seeds (in the mass range $10^4-10^6 M_{\odot}$) could both explain the
formation of massive black holes, but the abundance of seeds and their merging
mechanism are highly uncertain. In the next decades, the gravitational-wave
observatories coming online are expected to observe very high-redshift mergers,
shedding light on the seeding of the first black holes. In this Letter we
explore the potential and limitations for LISA, Cosmic Explorer and Einstein
Telescope to constrain the mixture ratio of light and heavy seeds as well as
the probability that central black holes in merging galaxies merge as well.
Since the third generation ground-based gravitational-wave detectors will only
observe light seed mergers, we demonstrate two scenarios in which the inference
of the seed mixture ratio and merging probability can be limited. The synergy
of multi-band gravitational-wave observations and electromagnetic observations
will likely be necessary in order to fully characterize the process of
high-redshift black hole formation.
