  We present the detection of multiple CO line transitions with ALMA in a few
tens of infrared-selected galaxies on and above the main sequence at z=1.1-1.7.
We reliably detected the emission of CO(5-4), CO(2-1), and CO(7-6)+[CI](2-1) in
50, 33, and 13 galaxies, respectively, and we complemented this information
with available CO(4-3) and [CI](1-0) fluxes for part of the sample, and
modeling of the optical-to-mm SEDs. We retrieve a quasi-linear relation between
LIR and CO(5-4) or CO(7-6) for main-sequence galaxies and starbursts,
corroborating the hypothesis that these transitions can be used as SFR tracers.
We find the CO excitation to steadily increase as a function of the star
formation efficiency, the mean intensity of the radiation field warming the
dust, the surface density of SFR, and, less distinctly, with the distance from
the main sequence. This adds to the tentative evidence for higher excitation of
the CO+[CI] SLED of starbursts relative to that for main-sequence objects,
where the dust opacities play a minor role in shaping the high-J CO transitions
in our sample. However, the distinction between the average SLED of upper
main-sequence and starburst galaxies is blurred, driven by a wide variety of
intrinsic shapes. LVG radiative transfer modeling demonstrates the existence of
a highly excited component that elevates the CO SLED of high-redshift
main-sequence and starbursting galaxies above the typical values observed in
the disk of the Milky Way. This excited component is dense and it encloses ~50%
of the total molecular gas mass in main-sequence objects. We interpret the
observed trends involving the CO excitation as mainly driven by a combination
of large SFRs and compact sizes, naturally connected with enhanced dense
molecular gas fractions and higher dust and gas temperatures, due to increasing
UV radiation fields, cosmic ray rates, and dust/gas coupling. [Abridged]
