  We present cosmological results inferred from the effective-field theory
(EFT) analysis of the full-shape of eBOSS quasars (QSO) power spectrum. We
validate our analysis pipeline against simulations, and find overall good
agreement between the analyses in Fourier and configuration space. Keeping the
baryon abundance and the spectral tilt fixed, we reconstruct at $68\%$ CL the
fractional matter abundance $\Omega_m$, the reduced Hubble constant $h$, and
the clustering amplitude $\sigma_8$, to respectively $\Omega_m=0.327\pm 0.035$,
$h=0.655\pm 0.034$, and $\sigma_8=0.880\pm 0.083$ from eBOSS QSO alone. These
constraints are consistent at $\lesssim 1.8\sigma$ with the ones from Planck
and from the EFT analysis of BOSS full-shape. Interestingly $S_8$ reconstructed
from eBOSS QSO is slightly higher than that deduced from Planck and BOSS,
although statistically consistent. In combination with the EFT likelihood of
BOSS, supernovae from Pantheon, and BAO from lyman-$\alpha$ and 6dF/MGS,
constraints improve to $\Omega_m = 0.2985\pm 0.0069$ and $h = 0.6803\pm
0.0075$, in agreement with Planck and with similar precision. We also explore
one-parameter extensions to $\Lambda$CDM and find that results are consistent
with flat $\Lambda$CDM at $\lesssim 1.3\sigma$. We obtain competitive
constraints on the curvature density fraction $\Omega_k=-0.039\pm 0.029$, the
dark energy equation of state $w_0=-1.038\pm 0.041$, the effective number of
relativistic species $N_{\rm eff}=3.44^{+0.44}_{-0.91}$ at $68\%$ CL, and the
sum of neutrino masses $\sum m_\nu<0.274e$V at $95\%$ CL, without Planck data.
Including Planck data, contraints significantly improve thanks to the large
lever arm in redshift between LSS and CMB measurements. In particular, we
obtain the stringent constraint $\sum m_\nu<0.093e$V, competitive with recent
lyman-$\alpha$ forest power spectrum bound.
