  Inclusive Higgs measurements at the LHC have limited resolution on the gluon
fusion loops, being unable to distinguish the long-distance contributions
mediated by the top quark from possible short-distance new physics effects.
Using an Effective Field Theory (EFT) approach we compare several proposed
methods to lift this degeneracy, including $t\bar{t}h$ and boosted, off-shell
and double Higgs production, and perform detailed projections to the
High-Luminosity LHC and a future hadron collider. In addition, we revisit
off-shell Higgs production. Firstly, we point out its sensitivity to
modifications of the top-$Z$ couplings, and by means of a general analysis we
show that the reach is comparable to that of tree-level processes such as
$t\bar{t}Z$ production. Implications for composite Higgs models are also
discussed. Secondly, we assess the regime of validity of the EFT, performing an
explicit comparison for a simple extension of the Standard Model containing one
vector-like quark.
