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As of this writing, travelers entering New York from one of the listed states must fill out the traveler health form (on paper if arriving by air or online if arriving by other means) and then quarantine for fourteen days upon arrival in New York. These rules apply equally to visitors and New York residents returning home.
It should be noted that the situation is subject to change between now and September, and states could be added or removed, or any other aspect of the program could be modified.