There is nothing in the Chinese Constitution about the separation of Church and State. On the other hand, of course, for a long time even a hint of any religious leaning could get people in trouble. That is changed. Nowadays there are a lot more people worshiping to different faiths. Not surprisingly, there is something of a hands-on approach taking by the Chinese government to that process. There was a blow up with the Catholic church a while ago as the Chinese demanded to approve appointments made from Rome - but things seem to be smooth now as both the Vatican and the Central Committee have agreed on the appointment of a new Bishop for Guangzhou (there had not been one for 6 years).
On a higher plain, it seems that the Dalai Lama can still easily irk the powers that be in Beijing. He has called for a referendum to determine whether and how the next Dalai Lama will be chosen and this has caused some consternation. Coming on the heels of the Congressional Gold Medal awarded to the Dalai Lama, that must really have stung. Beijing wants a say in who the next one will be; CNN says they are looking to recruit a pro-Beijing leader. I guess that would be the baby born with the iron rice-bowl in its mouth.