  The violation of the classical bounds imposed by Leggett-Garg inequalities
has tested the quantumness of neutrino oscillations (NOs) over a long distance
during the propagation. The measure of quantumness in experimentally observed
NOs is studied via quantum resource theory (QRT). Here, we focus on the
trade-off relations of QRT in the three-flavor NOs, based on Bell-type
violations, first-order coherence and intrinsic concurrence, and the relative
entropy of coherence. For the electron and muon antineutrino oscillations, the
analytical trade-off relations obeyed by the Bell-CHSH inequality of pairwise
flavor states in this three-flavor neutrino system are obtained; the sum of the
maximal violation of the CHSH tests for three pairwise flavor states is less
than or equal to 12. Moreover, there exists an equality relation concerning
first-order coherence and intrinsic concurrence in NOs, showing how much
quantum resources flow between first-order coherence and intrinsic concurrence
during the neutrino propagation. In addition, it is found that the tripartite
coherence of three-flavor system is equal to or larger than the sum of the
coherence of reduced bipartite flavor states. The trade-off relations of QRT
provide a method for studying how the quantum resources convert and distribute
in NOs, which might inspire the future applications in quantum information
processing using neutrinos.
