I got a pair of diamond earrings(worth $1600) for my wife while in India. My question is can I bring it back with me to the US as my personal possession or do I have to declare it at the customs?
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Yes, you should declare in Form 6059.
On the backside you have:
U.S. Residents — Declare all articles that you have acquired abroad and are bringing into the United States.
(... for non resident)
Declare all articles on this declaration form and show the value in U.S. dollars. For gifts, please indicate the retail value.
Duty — CBP officers will determine duty. U.S. residents are normally entitled to a duty-free exemption of $800 on items accompanying them. Visitors (non-residents) are normally entitled to an exemption of $100. Duty will be assessed at the current rate on the first $1,000 above the exemption.
So you should declare them, and pay duty on 800 USD.
You should have asked for duty free exemption on the Indian shop. Often (on touristic sites) they offer it to you.
BTW: you should be really sure that the diamonds are real: your wife known it a lot better. Or just declare them as fake diamonds, but tell the officer you have doubt, they will check (hopefully quickly) for you, so if they are fake you have not paid the duty.