Should you hide that you are already enrolled elsewhere when applying for another PhD? I plan to re-apply for a PhD in the US next year, but meanwhile, I decided to enroll in a PhD in my home country
I read the similar post: Is it a good idea to stay an extra year at my masters institution while re-applying to PhD programs elsewhere?

However, I am wondering whether you should admit/declare (in your application), that you are currently enrolled in a PhD program or it is better to hide this information from the US universities?
Thanks!
I answered a similar question here: No, you absolutely must not hide it.

The application will request your complete educational history, with official grades and records from each institution.  At the end, you will be asked to sign a statement that everything is true and complete.

Submitting an application that is false or has significant omissions is considered completely unethical.  It's very likely that the institution where you're applying will eventually find out, and there can be extremely serious consequences.

If they find out while reviewing your application, they'll reject it immediately, even if you would otherwise have been accepted.

If they accept you into the program and then find out while you're enrolled, they may kick you out of the program, even if you're doing well.  With a record of dishonesty, you will probably not get into another one.

If you enroll in the program and finish your PhD, and they find out about the false application years later, they can revoke your degree.  You will almost certainly be fired from whatever job you have at that time, and you will not work in academia again.

So you really do need to include your current institution in your application.  You'll probably also need a letter of recommendation from someone at your current institution.