  The statistical properties of spectra of quantum systems within the framework
of random matrix theory is widely used in many areas of physics. These
properties are affected, if two or more sets of spectra are superposed,
resulting from the discrete symmetries present in the system. Superposition of
spectra of $m$ such circular orthogonal, unitary and symplectic ensembles are
studied numerically using higher-order spacing ratios. For given $m$ and the
Dyson index $\beta$, the modified index $\beta'$ is tabulated whose nearest
neighbor spacing distribution is identical to that of $k$-th order spacing
ratio. For the case of $m=2$ ($m=3$) in COE (CUE) a scaling relation between
$\beta'$ and $k$ is given. For COE, it is conjectured that for $k=m+1$
($m\geq2$) and $k=m-3$-th ($m\geq5$) order spacing ratio distribution the
$\beta'$ is $m+2$ and $m-4$ respectively. Whereas in the case of CSE, for
$k=m+1$ ($m\geq2$) and $k=m-1$-th ($m\geq3$) the $\beta'$ is $2m+3$ and
$2(m-2)$ respectively. We also conjecture that for given $m$ ($k$) and $\beta$,
the sequence of $\beta'$ as a function of $k$ ($m$) is unique. Strong numerical
evidence in support of these results is presented. These results are tested on
complex systems like the measured nuclear resonances, quantum chaotic kicked
top and spin chains.
