  The proton-boron (p-$^{11}$B) reaction is regarded as the holy grail of
advanced fusion fuels, since the primary reaction produces three $\alpha$
particles with few neutrons and induced radio-activities from second order
reactions. Compared to the Deuterium-Tritium reaction a much higher reaction
temperature is required. Moreover, bremsstrahlung energy losses due to the high
nuclear charge of boron deem it seemingly apparent than a fusion reactor based
on Deuterium-Tritium plasma in equilibrium is to say the least very
difficult.It is becoming more appealing to collide intense laser beams or
accelerated proton beams with a boron target to produce p-$^{11}$B reactions.
The fusion yield of p-$^{11}$B reactions is closely related to proton beam
parameters and boron target conditions such as density, temperature, and
ingredients. Quantum degeneracy will increase fusion yields by reducing the
stopping power of injected protons. In this work, we suggest a high-efficiency
scheme for beam-target p-$^{11}$B fusions via injecting a MeV proton beam into
a highly compressed quantum degenerated boron target. Such a boron target can
be achieved via quasi-isentropic compression of solid boron by using precisely
shaped laser pulses. Our results indicate that for densities ranging from
$10^3$ to $10^4\rho_s$, where $\rho_s$ is the density of solid boron,
contributions of bound and free electrons to the stopping of protons can be
completely disregarded and dramatically reduced respectively. The result is an
increase in fusion yield by orders of magnitude. Furthermore, in order to
achieve multiplication factor $F$ greater than one, with $F$ defined as the
ratio of output fusion energy to the energy of injected protons, it is found
there exits a minimum possible density of boron target, which is $2.15 \times
10^4 \rho_s$ when the kinetic energy of injected protons is $0.8$ MeV.
