  Supersolid is an exotic quantum state of matter that hosts spontaneously the
features of both solid and superfluidity, which breaks the lattice
translational symmetry and U(1) gauge symmetry. Here we conduct inelastic
neutron scattering (INS) measurements and tensor-network calculations on the
triangular-lattice cobaltate Na$_2$BaCo(PO$_4$)$_2$, which is proposed in
[Xiang ${\it et al.}$, Nature 625, 270-275 (2024)] as a quantum magnetic analog
of supersolid. We uncover characteristic dynamical signatures, which include
distinct magnetic Bragg peaks indicating out-of-plane spin solidity and gapless
Goldstone modes corresponding to the in-plane spin superfluidity, offering
comprehensive spectroscopic evidence for spin supersolid in
Na$_2$BaCo(PO$_4$)$_2$. We also compute spin dynamics of the easy-axis
triangular-lattice model, and reveal magnon-roton excitations containing U(1)
Goldstone and roton modes associated with the in-plane spin superfluidity, as
well as pseudo-Goldstone and roton modes related to the out-of-plane spin
solidity, rendering double magnon-roton dispersions in the spin supersolid.
Akin to the role of phonon-roton dispersion in shaping the helium
thermodynamics, the intriguing magnetic excitations also strongly influence the
low-temperature thermodynamics of spin supersolid down to sub-Kelvin regime,
explaining the recently observed giant magnetocaloric effect in
Na$_2$BaCo(PO$_4$)$_2$.
