We report ALMA observations of the neutral atomic carbon transitions [CI] and multiple CO lines in a sample of $sim30$ main sequence galaxies at $zsim1$, including novel information on [CI](2-1) and CO(7-6) for 7 of such normal objects. We complement our observations with a collection of $>200$ galaxies with coverage of similar transitions, spanning the $z=0-4$ redshift interval and a variety of ambient conditions from local to high-redshift starbursts. We find systematic variations in the [CI]/IR and [CI]/high-$J$ ($J=7$) CO luminosity ratios among the various samples. We interpret these differences as increased dense molecular gas fractions and star formation efficiencies in the strongest high-redshift starbursts with respect to normal main sequence galaxies. We further report constant $L_{rm [CI]2-1}$/$L_{rm [CI]1-0}$ ratios across the galaxy populations and redshifts, suggesting that gas temperatures $T_{rm exc}$ traced by [CI] do not strongly vary. We find only a mild correlation with $T_{rm dust}$ and that, generally, $T_{rm exc} lesssim T_{rm dust}$. We fit the line ratios with classical PDR models, retrieving consistently larger densities and intensities of the UV radiation fields in submm galaxies than in main sequence and local objects. However, these simple models fall short in representing the complexity of a multiphase interstellar medium and should be treated with caution. Finally, we compare our observations with the Santa Cruz semi-analytical model of galaxy evolution, recently extended to simulate submm emission. While we confirm the success in reproducing the CO lines, we find systematically larger [CI] luminosities at fixed IR luminosity than predicted theoretically. This highlights the necessity of improving our understanding of the mechanisms regulating the [CI] emission on galactic scales. We release our data compilation to the community.
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