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 $Kge 3$. In order to study low-energy behaviors, we analyze these systems using t~Hooft anomaly matching and reliable semiclassics on $mathbb{R}^3times S^1$. 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 $mathbb{R}^3times S^1$ 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.
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