When did Constantine change the Roman state religion to Christianity?

Strictly speaking, Constantine did not make Christianity the state religion of Rome - this did not happen until 391 CE, some sixty years after his death.

However, Constantine did set in place the process by which Christianity would inevitably become the state religion. He gave Christianity state patronage and massive financial support, began the persecution of pagans, and stripped the pagan temples of their treasures, humiliating them in the eyes of the faithful. Nevertheless, at the time of his death, the majority of the population had remained faithful to the pagan gods and paganism remained nominally the state religion, or at least on an equal basis with Christianity.