  Fitting the thermal X-ray spectra of neutron stars (NSs) in quiescent X-ray
binaries can constrain the masses and radii of NSs. The effect of undetected
hot spots on the spectrum, and thus on the inferred NS mass and radius, has not
yet been explored for appropriate atmospheres and spectra. A hot spot would
harden the observed spectrum, so that spectral modeling tends to infer radii
that are too small. However, a hot spot may also produce detectable pulsations.
  We simulated the effects of a hot spot on the pulsed fraction and spectrum of
the quiescent NSs X5 and X7 in the globular cluster 47 Tucanae, using
appropriate spectra and beaming for hydrogen atmosphere models, incorporating
special and general relativistic effects, and sampling a range of system
angles. We searched for pulsations in archival Chandra HRC-S observations of X5
and X7, placing 90\% confidence upper limits on their pulsed fractions below
16\%. We use these pulsation limits to constrain the temperature differential
of any hot spots, and to then constrain the effects of possible hot spots on
the X-ray spectrum and the inferred radius from spectral fitting. We find that
hot spots below our pulsation limit could bias the spectroscopically inferred
radius downward by up to 28\%. For Cen X-4 (which has deeper published
pulsation searches), an undetected hot spot could bias its inferred radius
downward by up to 10\%. Improving constraints on pulsations from quiescent
LMXBs may be essential for progress in constraining their radii.
