0121 stock-picker has the "technical" answer right: All 50 states have "lost & found" laws, which require you to turn in any item, to the police, over a certain dollar amount. Like in CA (for instance) the criteria threshold is $100 value. The law does not state how this is valued. Eg.: the "melt" intrinsic value of an item? or the replacement cost? etc.... For example: the "intrinsic" scrap value of a brand new I-phone may be 30 cents (silicon, a few ounces of metals, etc..), yet obviously it's worth over $100 on ebay

Or a class ring might have only $70 scrap gold value, yet costs buyers at their school $300! (or whatever).
I ran this question by a lawyer friend of mine, and oddly, when posed with this ambiguity (how to value something), he said to just turn it in, and let the cops decide. ALSO the law makes no distinction about when YOU think something was lost. For example: if you found the ring 6" deep, indicating it was lost for over 50 yrs, (based on relative strata's of coin depths there), the law, again, makes no distinction of when YOU think it was "abandoned". Because, heck, then everyone could simply disregard the law and say "but I thought this giant gold ring was abandoned".
Yes, the L&F laws were born out of wandering cattle type things in the 1800s (so the neighbors couldn't simply say "I found it"). And are still logically applied to things like if the back door of a brinks armored car opens up, and wads of cash spill out (so passerbys can't simply say "I found it", and walk away). So the laws make sense, lest every thief caught with stolen merchandise could simply say "I found it", and think they had no obligation to seek out the owner (turn it in to the police).
But let's face it: None of us are doing it. A quick look at any beach hunters forum, and you will see nice rings posted all the time by md'rs in the gloating "today's finds" forums, right? But do you really think any of them are running down to the police station each time to turn it in?
In 30 days, if no one claims the item, the finder gets to keep it. But it's fraught with other issues though: Like in some states, if the value exceeds a certain dollar value ($250 in CA for example), the police will run a public notice in the local paper, inviting people to come down and identify and claim your property (with some sort of vague description, and I guess you have to come down and provide whatever proof that it's yours). But if the item is still not claimed by anyone, then the finder gets it. But get this: guess who has to pay for the cost of the newspaper found ad? YOU DO (if you intended to keep it after that 30 days).
And quite frankly, I'd be concerned that things you turn in to the cops would always have an owner "found", simply via ... well .... how do I put this .... "corruption". I mean, what's to stop someone there from simply picking up the phone, calling their uncle Fred, and saying: "Hey uncle Fred, do you want a nice Rolex? Come down to the station, and describe a rolex with these features, lost at such & such beach, and presto, you'll have yourself a nice new rolex". Sounds dastardly and evil, but think about it: In the eyes of the cops, it wasn't yours TO BEGIN WITH. You only "found" it. So what have you lost? You turned it in with the full expectation that it might be claimed.