  Verifying quantum systems has attracted a lot of interest in the last
decades. In this paper, we study the quantitative model-checking of quantum
continuous-time Markov chains (quantum CTMCs). The branching-time properties of
quantum CTMCs are specified by continuous stochastic logic (CSL), which is
famous for verifying real-time systems, including classical CTMCs. The core of
checking the CSL formulas lies in tackling multiphase until formulas. We
develop an algebraic method using proper projection, matrix exponentiation, and
definite integration to symbolically calculate the probability measures of path
formulas. Thus the decidability of CSL is established. To be efficient,
numerical methods are incorporated to guarantee that the time complexity is
polynomial in the encoding size of the input model and linear in the size of
the input formula. A running example of Apollonian networks is further provided
to demonstrate our method.
