  One proposed formation channel for stellar mass black holes (BHs) is through
hierarchical mergers of smaller BHs. Repeated mergers between comparable mass
BHs leave an imprint on the spin of the resulting BH, since the final BH spin
is largely determined by the orbital angular momentum of the binary. We find
that for stellar mass BHs forming hierarchically the distribution of spin
magnitudes is universal, with a peak at $a \sim 0.7$ and little support below
$a \sim 0.5$. We show that the spin distribution is robust against changes to
the mass ratio of the merging binaries, the initial spin distribution of the
first generation of BHs, and the number of merger generations. While we assume
an isotropic distribution of initial spin directions, spins that are
preferentially aligned or antialigned do not qualitatively change our results.
We also consider a "cluster catastrophe" model for BH formation in which we
allow for mergers of arbitrary mass ratios and show that this scenario predicts
a unique spin distribution that is similar to the universal distribution
derived for major majors. We explore the ability of spin measurements from
ground-based gravitational-wave (GW) detectors to constrain hierarchical merger
scenarios. We apply a hierarchical Bayesian mixture model to mock GW data and
argue that the fraction of BHs that formed through hierarchical mergers will be
constrained with $\mathcal{O}(100)$ LIGO binary black hole detections, while
with $\mathcal{O}(10)$ detections we could falsify a model in which all
component BHs form hierarchically.
