  We study constraints on primordial mode-coupling from the power spectrum,
squeezed-limit bispectrum and collapsed trispectrum of matter and halos. We
describe these statistics in terms of long-wavelength $2$-point functions
involving the matter/halo density and position-dependent power spectrum. This
allows us to derive simple, analytic expression for the information content,
treating constraints from scale-dependent bias in the halo power spectrum on
the same footing as those from higher order statistics. In particular, we
include non-Gaussian covariance due to long-short mode-coupling from non-linear
evolution, which manifests itself as long-mode cosmic variance in the
position-dependent power spectrum. We find that bispectrum forecasts that
ignore this cosmic variance may underestimate $\sigma(f_{\rm NL})$ by up to a
factor $\sim 3$ for the matter density (at $z=1$) and commonly a factor $\sim
2$ for the halo bispectrum. Constraints from the bispectrum can be improved by
combining it with the power spectrum and trispectrum. The reason is that, in
the position-dependent power spectrum picture, the bispectrum and trispectrum
intrinsically incorporate multitracer cosmic variance cancellation, which is
optimized in a joint analysis. For halo statistics, we discuss the roles of
scale-dependent bias, matter mode-coupling, and non-linear, non-Gaussian
biasing ($b_{11}^{(h)}$). While scale-dependent bias in the halo power spectrum
is already very constraining, higher order halo statistics are competitive in
the regime where stochastic noise in the position-dependent halo power spectrum
is low enough for cosmic variance cancellation to be effective, i.e.~for large
halo number density and large $k_{\rm max}$. This motivates exploring this
regime observationally.
