  We study low-energy dynamics of $[SU(N)]^K$ chiral quiver gauge theories in
connection with $\mathcal{N}=1$ super Yang-Mills (SYM) theory, and quantum
chromodynamics with bi-fundamental fermions (QCD(BF)). These theories can be
obtained by $\mathbb{Z}_K$ orbifold projections of $\mathcal{N}=1$ $SU(NK)$ SYM
theory, but the perturbative planar equivalence does not extend
nonperturbatively for $K\ge 3$. In order to study low-energy behaviors, we
analyze these systems using 't~Hooft anomaly matching and reliable semiclassics
on $\mathbb{R}^3\times S^1$. Thanks to 't~Hooft anomaly that involves $1$-form
center symmetry and discrete chiral symmetry, we predict that chiral symmetry
must be spontaneously broken in the confinement phase, and there exist $N$
vacua. Theories with even $K$ possess a physical $\theta$ angle despite the
presence of massless fermions, and we further predict the $N$-branch structure
associated with it; the number of vacua is enhanced to $2N$ at $\theta=\pi$ due
to spontaneous $CP$ breaking. Both of these predictions are explicitly
confirmed by reliable semiclassics on $\mathbb{R}^3\times S^1$ with the
double-trace deformation. Symmetry and anomaly of odd-$K$ theories are the same
as those of the ${\cal N}=1$ SYM, and the ones of even-$K$ theories are same as
those of QCD(BF). We unveil why there exists universality between vector-like
and chiral quiver theories, and conjecture that their ground states can be
continuously deformed without quantum phase transitions. We briefly discuss
anomaly inflow on the domain walls connecting the vacua of the theory and
possible anomaly matching scenarios.
