  Single-field models of $\alpha$-attractor quintessential inflation provide a
unified picture of the two periods of early- and late-time cosmic acceleration,
where both inflation and dark energy are described by a single scalar degree of
freedom rolling down a runaway potential. These theoretically well-motivated
models have distinct observational predictions that are in agreement with
existing cosmological data. We show that the next generation of large-scale
structure surveys, even when no other cosmological data sets are considered,
will strongly constrain the parameter space of these models, and test them
against the standard cosmological model and more conventional
non-quintessential inflation. In particular, we expect
$\mathcal{O}(10^{-5}\mathrm{-}10^{-4})$ constraints on the present values of
the dark energy equation of state and its time derivative, $w_0$ and $w_a$. We
also forecast more than one order of magnitude tighter constraints on the
spectral index of primordial curvature perturbations $n_s$ compared to the
expectations for the standard model. This demonstrates the powerful synergy
between the upcoming large-scale structure probes of inflation and those aiming
to measure the tensor-to-scalar ratio $r$ through the observation of $B$-mode
polarization of the cosmic microwave background.
