This is a follow-on question from Politics SE, wherein several participants claimed that it was illegal for a voter to divulge their own ballot choices. As this is purely a question of law, rather than politics, this issue is more appropriate for Law SE.
So, are there any US States that have made it illegal for a voter to disclose their ballot (voting) choices?
Short answer: It depends on the state and exactly how you do so. Stating how you voted, by itself, is fine; however, taking a photo of your ballot instead of just saying how you voted is illegal in some states, especially if the photo was taken within a polling place. Laws banning these so-called "ballot selfies" may be unconstitutional, and have been successfully challenged under the first amendment in some cases.
There is not a general ban on simply saying who you voted for. This is an extremely common practice, and it is the basis for "exit polling," where voters leaving a polling place are asked whom they voted for in order to collect voting statistics to predict the winner of the election. This type of polling has been upheld by courts, and it sounds like laws banning it were primarily concerned with voter intimidation rather than vote-buying.
As of 2016, 18 states banned sharing any photograph of a ballot, while an additional 6 ban photography in polling places. The rationale for such laws is to prevent vote-buying, because if you can't take a photo of your ballot, you can't prove whom you voted for (whereas without the photo, you could simply vote however you like, then lie). However, at least one such law has been challenged in federal court and invalidated as a violation of the first amendment (in Rideout v. Gardner) "because it is a content-based restriction on speech that cannot survive strict scrutiny." If such first-amendment challenges continue to be upheld, it is possible that this practice will be legal throughout the United States.
I don't know of any law that prohibits the disclosure of your choices on a ballot. It would certainly be invalidated on First Amendment grounds.
There are laws outlawing the disclosure of pictures of your ballot, but there is little remaining debate that these laws violate the First Amendment. New Hampshire's ballot-selfie law was invalidated in 2015, and the First Circuit Court of Appeals went a step further in 2016, making it illegal to enforce such a law anywhere else in the circuit:
Even accepting the possibility that ballot selfies will make vote buying and voter coercion easier by providing proof of how the voter actually voted, the statute still fails for lack of narrow tailoring. “[B]y demanding a close fit between ends and means, the tailoring requirement [under intermediate scrutiny] prevents the government from too readily ‘sacrific[ing] speech for efficiency.’ ” ...
New Hampshire has “too readily forgone options that could serve its interests just as well, without substantially burdening” legitimate political speech. At least two different reasons show that New Hampshire has not attempted to tailor its solution to the potential problem it perceives. First, the prohibition on ballot selfies reaches and curtails the speech rights of all voters, not just those motivated to cast a particular vote for illegal reasons. ... Second, the State has not demonstrated that other state and federal laws prohibiting vote corruption are not already adequate to the justifications it has identified. ... New Hampshire suggests that it has no criminal statute preventing a voter from selling votes. That can be easily remedied without the far reach of this statute. The State may outlaw coercion or the buying or selling of votes without the need for this prohibition.
Rideout v. Gardner, 838 F.3d 65, 74 (1st Cir. 2016)
Michigan has agreed to stop enforcing its ballot selfie law, and Indiana has been permanently enjoined from enforcing its law. New York did survive a challenge to its law, but that one is generally considered an outlier.
Since the mod welcomes answers from other jurisdictions a short note on the situation in Germany.
In 2017, the head of the federal election commission ("Bundeswahlleiter") filed charges against several persons who took selfies of their mail-in ballots in their homes and published them.
He referred to §107c of the penal code that forbids to violate regulations serving the secrecy of the election by disclosing how somebody voted. He argued that "somebody" includes oneself, an opinion not shared by the prosecutor who consequently dropped the charges. A link to that story, in German, is here.
Taking pictures in the voting booth is unmistakably forbidden though.
In the UK, it's a crime to take any photographs or images inside the polling station. Photographing a ballot paper is a big no-no. Although prosecutions to my knowledge are extremely rare and you would realistically probably just get a telling-off from the police. Unless you did something really egregious like walk around a polling station photographing all the ballot papers in sight and posting them on Twitter or something as crazy as that.
Of course, you're allowed to tell people how you voted.
