Connecticut has not legalized marijuana. Possession of under 1/2 oz has been decriminalized, but still carries a heft fine. Possession of larger amounts will likely lead to jail time. They have a medical marijuana program which started in 2012. You need a card which is issued by doctor to be a part of the program. There are a number of qualifying medical conditions that make one eligible for the program. It is very difficult to obtain one of these cards. There are no laws exempting tourists from the general penalties for marijuana possession.
Connecticut does not honor other states medical marijuana cards. You must obtain a Connecticut prescription or card in order to legally purchase marijuana in the state of Connecticut. Otherwise marijuana possession of up to a half ounce is a civil penalty and 150$ fine. This applies for anyone, locals and visitors alike.
The laws are the same regardless whether or not you’re a tourist or a traveler. Check the ordinances of the part of Connecticut you will be visiting for more specific details on a particular marijuana policy. In most states, possessing marijuana is illegal. If you are caught with the substance you may face either a civil fine or depending on how much you have, be arrested and given a criminal charge.