  The random matrix theory (RMT) can be used to classify both topological
phases of matter and quantum chaos. We develop a systematic and transformative
RMT to classify the quantum chaos in the colored Sachdev-Ye-Kitaev (SYK) model
first introduced by Gross and Rosenhaus. Here we focus on the 2-colored case
and 4-colored case with balanced number of Majorana fermion $N$. By identifying
the maximal symmetries, the independent parity conservation sectors, the
minimum (irreducible) Hilbert space, and especially the relevant anti-unitary
and unitary operators, we show that the color degree of freedoms lead to novel
quantum chaotic behaviours. When $N$ is odd, different symmetry operators need
to be constructed to make the classifications complete. The 2-colored case only
show 3-fold Wigner-Dyson way, and the 4-colored case show 10-fold generalized
Wigner-Dyson way which may also have non-trivial edge exponents. We also study
2- and 4-colored hybrid SYK models which display many salient quantum chaotic
features hidden in the corresponding pure SYK models. These features motivate
us to develop a systematic RMT to study the energy level statistics of 2 or 4
un-correlated random matrix ensembles. The exact diagonalizations are performed
to study both the bulk energy level statistics and the edge exponents and find
excellent agreements with our exact maximal symmetry classifications. Our
complete and systematic methods can be easily extended to study the generic
imbalanced cases. They may be transferred to the classifications of colored
tensor models, quantum chromodynamics with pairings across different colors,
quantum black holes and interacting symmetry protected (or enriched)
topological phases.
