  Future surveys of large-scale structure will be able to measure perturbations
on the scale of the cosmological horizon, and so could potentially probe a
number of novel relativistic effects that are negligibly small on sub-horizon
scales. These effects leave distinctive signatures in the power spectra of
clustering observables and, if measurable, would open a new window on
relativistic cosmology. We quantify the size and detectability of the effects
for the most relevant future large-scale structure experiments: spectroscopic
and photometric galaxy redshift surveys, intensity mapping surveys of neutral
hydrogen, and radio continuum surveys. Our forecasts show that next-generation
experiments, reaching out to redshifts $z\simeq 4$, will not be able to detect
previously-undetected general-relativistic effects by using individual tracers
of the density field, although the contribution of weak lensing magnification
on large scales should be clearly detectable. We also perform a rigorous joint
forecast for the detection of primordial non-Gaussianity through the excess
power it produces in the clustering of biased tracers on large scales, finding
that uncertainties of $\sigma(f_{\rm NL})\sim 1-2$ should be achievable. We
study the level of degeneracy of these large-scale effects with several
tracer-dependent nuisance parameters, quantifying the minimal priors on the
latter that are needed for an optimal measurement of the former. Finally, we
discuss the systematic effects that must be mitigated to achieve this level of
sensitivity, and some alternative approaches that should help to improve the
constraints. The computational tools developed to carry out this study, which
requires the full-sky computation of the theoretical angular power spectra for
$\mathcal{O}(100)$ redshift bins, as well as realistic models of the luminosity
function, are publicly available.
