  Recently, it has been shown that the quantum Fisher information via local
observables and via local measurements (i.e., local quantum Fisher information
(LQFI)) is a central concept in quantum estimation and quantum metrology and
captures the quantumness of correlations in the multi-component quantum system
[S. Kim et al., Phys. Rev. A. 97, 032326 (2018)]. This new discord-like measure
is very similar to the quantum correlations measure called local quantum
uncertainty (LQU). In the present study, we have revealed that LQU is bounded
by LQFI in the phase estimation protocol. Also, a comparative study between
these two quantum correlations quantifiers is addressed for the quantum
Heisenberg $XY$ model. Two distinct situations are considered. The first one
concerns the anisotropic $XY$ model and the second situation concerns the
isotropic $XY$ model submitted to an external magnetic field. Our results
confirm that LQFI reveals more quantum correlations than LQU.
