Does prison time erase sins in Islam?

Hardship can wipe away sins

The Prophet ﷺ said that no tiredness, illness, worry, grief, harm or sadness reaches a Muslim, not even a thorn's prick, without Allah removing some of his sins because of it (Sahih al-Bukhari 5641). Losing your freedom, being away from family and living with regret are real hardships. If you face them with patience and turn to Allah, they are not wasted.

Worldly punishment and the Hereafter

When the Companions pledged to avoid major sins, the Prophet ﷺ told them that whoever committed one and was punished for it in this world, that punishment would be an expiation for him (Sahih al-Bukhari 18). Scholars mostly applied this to the fixed hudud penalties of Islamic law. Whether a prison sentence under a secular system works the same way is discussed among scholars, and many say it doesn't replace repentance. See What are hudud punishments in Islam?

Repentance is the real key

Allah's door is open to anyone who turns back to Him, whatever they did:

Sincere repentance means regret, stopping the sin and resolving not to return to it. Good deeds after that help too. See Can good deeds erase bad deeds? And if you embraced Islam in prison, your past sins are wiped clean by your shahada. See Are all my sins forgiven when I become Muslim?

Rights of other people

Sins against people, like theft, fraud or violence, need one more step: making it right with them where you can. The Prophet ﷺ described the truly bankrupt person as someone who arrives on the Day of Judgement with prayer and fasting, but also with wrongs done to others, so his good deeds are handed to them (Sahih Muslim 2581). Paying back money, apologizing when it's safe, and making dua for those you harmed all count.