We know that the Roman Empire became a Christian state. Was it illegal to blaspheme God in Antioch in 360.-410.? What punishment was prescribed (if it was illegal)? If it was punishable, who could carry out the punishment?
Edit.
So pardon for my rough approximation in the original post, I would really like to know the law which governed blasphemy (of Christ) in Antioch in the period of 386.-397.
Well, if we can change venue to Alexandria, which was a Roman city in Egypt with roughly similar standing to Antioch (they both housed a Christian Patriarch), and roll the date forward by only 5 years, then the fate of Hypatia might be a pretty good guide.
The short version is that she was a pagan philosopher, who was well-liked in the pagan community, and was a close friend and advisor to the Roman Prefect. When the Prefect got into a political dispute with the new Bishop of Alexandria, she became a target. Eventually a Christian mob captured her and ... well did about all the disgusting things you can imagine a mob doing to a target, and then some.
What this incident tells us:
Being pagan (and teaching philosophy that wasn't Christian) was not against the law. Not only was she able to do that openly in Alexandria, but it didn't stop her from being pals with the person primarily in charge of enforcing Roman Law in the city. Additionally, she wasn't the only one. There was an entire community of Pagans.
We have a pretty good idea she was accused of all kinds of blasphemous things, not only because that's what would be required to rile a Christian crowd into a lynch mob, but because there seems to be a memory of them from the later writings of Bishop John of Niku:
and she was devoted at all times to magic, astrolabes and instruments of music, and she beguiled many people through her Satanic wiles. And the governor of the city honored her exceedingly; for she had beguiled him through her magic.
Apparently these charges were not something one could take to the secular authorities to get someone arrested, or the mob would not have been nessecary. So it wasn't against the law.
One might note from the question's Chrysostom passage that he specifically suggested listeners take physical (violent) actions themselves when they see such a "crime", which also indicates that simply having the person antiseptically arrested and handed over to authorities is not actually an option.
Anyone from a minority community can tell you there's often a large difference between what's technically legal for them to do, and what the majority will actually allow them to do. Hypatia would probably tell you the same, if she could.
In 361-363 the Empire was ruled by a pagan emperor. His successors were Christian, but the empire was still a multi-religious state. It is only in 381 (under Theodosius) that introduction of uniformity and persecution of non-Christians began. So you can expect blasphemy laws from that time only and they were gradually introduced.
The citation from John Chrysostom that you give supports this: he calls to "rebuke and inflict blows" rather than "denounce to the authorities". Which makes it clear that blasphemy was not illegal at the time he wrote this.
To blaspheme which god? In 363, if you blasphemed the pagan gods, an emperor wrote disparagingly about your city. If you blasphemed Christ, the Persian army slew you at Samarra.
This doesn't apply to the other years. It's interesting that you included Julian's reign.
