  In July of 2019, the IceCube experiment detected a high-energy neutrino from
the direction of the powerful blazar PKS 1502+106. We perform multi-wavelength
and multi-messenger modeling of this source, using a fully self-consistent
one-zone model that includes the contribution of external radiation fields
typical of flat-spectrum radio quasars (FSRQs). We identify three different
activity states of the blazar: one quiescent state and two flaring states with
hard and soft gamma-ray spectra. We find two hadronic models that can describe
the multi-wavelength emission during all three states: a leptohadronic model
with a contribution from photo-hadronic processes to X-rays and gamma rays, and
a proton synchrotron model, where the emission from keV to 10 GeV comes from
proton synchrotron radiation. Both models predict a substantial neutrino flux
that is correlated with the gamma-ray and soft X-ray fluxes. Our results are
compatible with the detection of a neutrino during the quiescent state, based
on event rate statistics. We conclude that the soft X-ray spectra observed
during bright flares strongly suggest a hadronic contribution, which can be
interpreted as additional evidence for cosmic ray acceleration in the source
independently of neutrino observations. We find that more arguments can be made
in favor of the leptohadronic model vis-a-vis the proton synchrotron scenario,
such as a lower energetic demand during the quiescent state. However, the same
leptohadronic model would be disfavored for flaring states of PKS 1502+106 if
no IceCube events were found from the direction of the source before 2010,
which would require an archival search.
