  For the foreseeable future, the exploration of the high-energy frontier will
be the domain of the Large Hadron Collider (LHC). Of particular significance
will be its high-luminosity upgrade (HL-LHC), which will operate until the
mid-2030s. In this endeavour, for the full exploitation of the HL-LHC physics
potential an improved understanding of the parton distribution functions (PDFs)
of the proton is critical. The HL-LHC program would be uniquely complemented by
the proposed Large Hadron electron Collider (LHeC), a high-energy lepton-proton
and lepton-nucleus collider based at CERN. In this work, we build on our recent
PDF projections for the HL-LHC to assess the constraining power of the LHeC
measurements of inclusive and heavy quark structure functions. We find that the
impact of the LHeC would be significant, reducing PDF uncertainties by up to an
order of magnitude in comparison to state-of-the-art global fits. In comparison
to the HL-LHC projections, the PDF constraints from the LHeC are in general
more significant for small and intermediate values of the momentum fraction x.
At higher values of x, the impact of the LHeC and HL-LHC data is expected to be
of a comparable size, with the HL-LHC constraints being more competitive in
some cases, and the LHeC ones in others. Our results illustrate the encouraging
complementarity of the HL-LHC and the LHeC in terms of charting the quark and
gluon structure of the proton.
