Transitioning from Stage-III to Stage-IV: Cosmology from
  galaxy$\times$CMB lensing and shear$\times$CMB lensing
  We examine the cosmological constraining power from two cross-correlation
probes between galaxy and CMB surveys: the cross-correlation of lens galaxy
density with CMB lensing convergence $\langle\delta_g\kappa_{\rm CMB}\rangle$
and source galaxy weak lensing shear with CMB lensing convergence
$\langle\gamma\kappa_{\rm CMB}\rangle$. These two cross-correlation probes
provide an independent cross-check of other large-scale structure constraints
and are insensitive to systematic effects that could be present in galaxy-only
or CMB-only analyses. We study how the constraining power of
$\langle\delta_g\kappa_{\rm CMB}\rangle+\langle\gamma\kappa_{\rm CMB}\rangle$
changes as the galaxy samples and CMB dataset qualitatively change from
Stage-III (ongoing) to Stage-IV (future) surveys. Given the flexibility in
selecting the lens galaxy sample, we also explore the impact on cosmological
constraints when we vary the redshift range and magnitude limit of the lens
galaxies using mock galaxy catalogs. We find that in our setup, cosmological
constraints from $\langle\delta_g\kappa_{\rm CMB}\rangle$ and
$\langle\gamma\kappa_{\rm CMB}\rangle$ are comparable in Stage-III; but as we
move to Stage-IV, shot noise from the galaxy density becomes subdominant to
cosmic variance, preventing the contribution from $\langle\delta_g\kappa_{\rm
CMB}\rangle$ to further improve. This implies that to maximize the cosmological
constraints from future $\langle\delta_g\kappa_{\rm CMB}\rangle+\langle
\gamma\kappa_{\rm CMB}\rangle$ analyses, we should focus more on the
requirements on $\langle\gamma \kappa_{\rm CMB}\rangle$ instead of
$\langle\delta_g\kappa_{\rm CMB}\rangle$. In addition, the selection of the
lens galaxy sample in $\langle\delta_g\kappa_{\rm CMB}\rangle$ should be
optimized in terms of our ability to characterize its redshift or galaxy bias
instead of its number density.
