  We review the prospects for quarkonium-production studies in proton and
nuclear collisions accessible during the upcoming phases of the CERN Large
Hadron Collider operation after 2021, including the ultimate high-luminosity
phase, with increased luminosities compared to LHC Runs 1 and 2. We address the
current experimental and theoretical open issues in the field and the
perspectives for future studies in quarkonium-related physics through the
exploitation of the huge data samples to be collected in proton-proton, with
integrated luminosities reaching up to 3/ab, in proton-nucleus and in
nucleus-nucleus collisions, both in the collider and fixed-target modes. Such
investigations include, among others, those of: (i) the quarkonia produced in
association with other hard particles; (ii) the chi(Q) and eta(Q) down to small
transverse momenta; (iii) the constraints brought in by quarkonia on gluon
PDFs, nuclear PDFs, TMDs, GPDs and GTMDs, as well as on the low-x parton
dynamics; (iv) the gluon Sivers effect in polarised-nucleon collisions; (v) the
properties of the quark-gluon plasma produced in ultra-relativistic heavy-ion
collisions and of collective partonic effects in general; and (vi) double and
triple parton scatterings.
