  Dense high-energy monoenergetic proton beams are vital for wide applications,
thus modern laser-plasma-based ion acceleration methods are aiming to obtain
high-energy proton beams with energy spread as low as possible. In this work,
we put forward a quantum radiative compression method to post-compress a highly
accelerated proton beam and convert it to a dense quasi-monoenergetic one. We
find that when the relativistic plasma produced by radiation pressure
acceleration collides head-on with an ultraintense laser beam, large-amplitude
plasma oscillations are excited due to quantum radiation-reaction and the
ponderomotive force, which induce compression of the phase space of protons
located in its acceleration phase with negative gradient. Our three-dimensional
spin-resolved QED particle-in-cell simulations show that hollow-structure
proton beams with a peak energy $\sim$ GeV, relative energy spread of few
percents and number $N_p\sim10^{10}$ (or $N_p\sim 10^9$ with a $1\%$ energy
spread) can be produced in near future laser facilities, which may fulfill the
requirements of important applications, such as, for radiography of ultra-thick
dense materials, or as injectors of hadron colliders.
