  The violation of Leggett-Garg inequalities tested the quantumness of neutrino
oscillations (NOs) across macroscopic distances. The quantumness can be
quantified by using the tools of the quantum resource theories. Recently, a new
genuine tripartite entanglement measure [S. B. Xie et al., Phys. Rev. Lett.
127, 040403 (2021)], concurrence fill, is defined as the square root of the
area of the concurrence triangle satisfying all genuine multipartite
entanglement conditions. It has several advantages compared to other existing
tripartite measures. Here, we focus on using concurrence fill to quantify the
tripartite entanglement in three-flavor NOs. Concurrence fill can reach its
maximum $0.89$ for the experimentally-observed electron antineutrino
oscillations, but it cannot for the muon antineutrino oscillations. In both
cases, we compare its performance with other three tripartite entanglement
measures, including the generalized geometric measure (GGM), the three-$\pi$
entanglement, and the genuinely multipartite concurrence (GMC), in the neutrino
propagation, and accordingly show that concurrence fill contains the most
quantum resource. Furthermore, concurrence fill and the three-$\pi$
entanglement are always smooth, while GGM and GMC measures have several sharp
peaks. The genuine tripartite quantification of the quantumness of three-flavor
NOs represents the first step towards the further potential application of
neutrinos on quantum information processing.
