In inertial confinement fusion (ICF), laser-plasma instability (LPI) induced by highly coherent light is a core bottleneck restricting high-gain energy yield, and spatiotemporal low-coherence lasers are recognized as an effective technical approach to mitigate this effect. Based on the physical mechanism of transverse mode degeneracy in a 4F imaging degenerate cavity, combined with the random complex screen method and approximate laser amplification dynamics equations, this paper constructs a theoretical model governing the nonlinear amplification propagation of spatiotemporal low-coherence lasers and the evolution of cross-spectral density, which can quantitatively characterize the gain and coherence evolution laws of broadband multi-transverse-mode optical fields. On this basis, utilizing the spatiotemporal low-coherence light generated by a 4F degenerate cavity Nd:glass laser as the seed source, experimental investigations on the amplification characteristics are systematically conducted on a multi-stage rod Nd:glass amplification platform. The efficient amplification of a spatiotemporal low-coherence laser with a pulse width of 6.92 ns is successfully demonstrated for the first time, achieving a maximum output energy of 52J and a net gain of 1.38×10⁵. Furthermore, the evolution laws of gain saturation and spectral narrowing during the amplification process are quantitatively revealed, with the spectral full width at half maximum (FWHM) narrowed from 3.86 nm to 1.77 nm, and the contribution proportion of the gain-bandwidth filtering effect is explicitly identified. The results confirm that the laser beam well preserves its initial low spatial coherence characteristics even after multi-stage high-gain amplification, as evidenced by the hard-edge diffraction fringe contrast dropping to 0.02, showing excellent agreement between the theoretical simulations and experimental data. This study provides a significant experimental foundation and critical theoretical support for the design of next-generation low-coherence laser drivers targeted at high-gain laser fusion.