  Investigation of dynamical excitations is difficult but crucial to the
understanding of many exotic quantum phenomena discovered in quantum materials.
This is particularly true for highly frustrated quantum antiferromagnets whose
dynamical properties deviate strongly from theoretical predictions made based
on the spin-wave or other approximations. Here we present a large-scale
numerical calculation on the dynamical correlation functions of spin-$1/2$
triangular Heisenberg model using a state-of-the-art tensor network
renormalization group method. The calculated results allow us to gain for the
first time a comprehensive picture on the nature of spin excitation spectra in
this highly frustrated quantum system. It provides a quantitative account for
all the key features of the dynamical spectra disclosed by inelastic neutron
scattering measurements for $\rm Ba_3CoSb_2O_9$, revealing the importance of
the interplay between low- and high-energy excitations and its renormalization
effect to the low-energy magnon bands and high-energy continuums. We identify
the longitudinal Higgs modes in the intermediate-energy scale and predict the
energy and momentum dependence of spectral functions along the three principal
axes that can be verified by polarized neutron scattering experiments.
Furthermore, we find that the spin excitation spectra weakly depend on the
anisotropic ratio of the antiferromagnetic interaction.
