  In this work, we conduct a detailed comparison between two second-order
conservative Allen-Cahn (AC) models [\emph{Model A}: Zheng \emph{et al.}, Phys.
Rev. E 101, 0433202 (2020) and \emph{Model B}: Mirjalili and Mani, (2023)] for
the immiscible $N$-phase flows. Mathematically, these two AC equations can be
proved to be equivalent under some approximate conditions. However, the effects
of these approximations are unclear from the theoretical point of view, and
would be considered numerically. To this end, we propose a consistent and
conservative lattice Boltzmann method for the AC models for $N$-phase flows,
and present some numerical comparisons of accuracy and stability between these
two AC models. The results show that both two AC models have good performances
in accuracy, but the \emph{Model B} is more stable for the realistic complex
$N$-phase flows, although there is an adjustable parameter in the \emph{Model
A}.
