  Light bosonic fields are ubiquitous in extensions of the Standard Model. Even
when minimally coupled to gravity, these fields might evade the assumptions of
the black-hole no-hair theorems and give rise to spinning black holes which can
be drastically different from the Kerr metric. Furthermore, they allow for
self-gravitating compact solitons, known as (scalar or Proca) boson stars. The
quasi-periodic oscillations (QPOs) observed in the X-ray flux emitted by
accreting compact objects carry information about the strong-field region, thus
providing a powerful tool to constrain deviations from Kerr's geometry and to
search for exotic compact objects. By using the relativistic precession model
as a proxy to interpret the QPOs in terms of geodesic frequencies, we
investigate how the QPO frequencies could be used to test the no-hair theorem
and the existence of light bosonic fields near accreting compact objects. We
show that a detection of two QPO triplets with current sensitivity can already
constrain these models, and that the future eXTP mission or a LOFT-like mission
can set very stringent constraints on black holes with bosonic hair and on
(scalar or Proca) boson stars. The peculiar geodesic structure of compact
scalar/Proca boson stars implies that these objects can easily be ruled out as
alternative models for X-ray source GRO J1655-40.
