  One of the central goals of the physics program at the future colliders is to
elucidate the origin of electroweak symmetry breaking, including precision
measurements of the Higgs sector. This includes a detailed study of Higgs boson
(H) pair production, which can reveal the H self-coupling. Since the discovery
of the Higgs boson, a large campaign of measurements of the properties of the
Higgs boson has begun and many new ideas have emerged during the completion of
this program. One such idea is the use of highly boosted and merged hadronic
decays of the Higgs boson ($\mathrm{H}\to\mathrm{b}\bar{\mathrm{b}}$,
$\mathrm{H}\to\mathrm{W}\mathrm{W}\to\mathrm{q}\bar{\mathrm{q}}\mathrm{q}\bar{\mathrm{q}}$)
with machine learning methods to improve the signal-to-background
discrimination. In this white paper, we champion the use of these modes to
boost the sensitivity of future collider physics programs to Higgs boson pair
production, the Higgs self-coupling, and Higgs-vector boson couplings. We
demonstrate the potential improvement possible at the Future Circular Collider
in hadron mode, especially with the use of graph neural networks.
