Question:

New Hampshire RSA 651:5 reads, in part::

X. Upon entry of an order of annulment:
       (a) The person whose record is annulled shall be treated in all respects as if he or she had never been arrested, convicted or sentenced, except that, upon conviction of any crime committed after the order of annulment has been entered, the prior conviction may be considered by the court in determining the sentence to be imposed, and may be counted toward habitual offender status under RSA 259:39.
       (b) The court shall issue the person a certificate stating that such person's behavior after the conviction has warranted the issuance of the order, and that its effect is to annul the arrest, conviction, and sentence, and shall notify the state police criminal records unit, the prosecuting agency, and the arresting agency.
       (c) The court records relating to an annulled arrest, conviction, or sentence shall be sealed and available only to the person whose record was annulled, his or her attorney, a court for sentencing pursuant to subparagraph (a), law enforcement personnel for legitimate law enforcement purposes, or as otherwise provided in this section.
       (d) Upon payment of a fee not to exceed $100 to the state police, the state police criminal records unit shall remove the annulled criminal record and inform all appropriate state and federal agencies of the annulment.
       (e) The arresting agency and the prosecuting agency shall clearly identify in their respective files and in their respective electronic records that the arrest or conviction and sentence have been annulled.
       (f) In any application for employment, license or other civil right or privilege, or in any appearance as a witness in any proceeding or hearing, a person may be questioned about a previous criminal record only in terms such as "Have you ever been arrested for or convicted of a crime that has not been annulled by a court?''
    XI. Nothing in this section shall affect any right:
       (a) Of the person whose record has been annulled to appeal from the conviction or sentence or to rely on it in bar of any subsequent proceedings for the same offense; or
       (b) Of law enforcement officers to maintain arrest and conviction records and to communicate information regarding the annulled record of arrest or conviction to other law enforcement officers for legitimate investigative purposes or in defense of any civil suit arising out of the facts of the arrest, or to the police standards and training council solely for the purpose of assisting the council in determining the fitness of an individual to serve as a law enforcement officer, in any of which cases such information shall not be disclosed to any other person.

CT Pistol Permit Application reads, in part:

With regard to criminal history information arising from jurisdictions other than the State of Connecticut: You are not required to disclose the existence of any arrest, criminal charge or conviction, the records of which have been erased pursuant to the law of the other jurisdiction. Additionally, you are not required to disclose the existence of an arrest arising from another jurisdiction if you are permitted under the law of that jurisdiction to swear under oath that you have never been arrested.
Have you ever been CONVICTED under the laws of this state, federal law or the laws of another jurisdiction? NO YES If "YES,” list all convictions, include charges, location, date of arrest, and disposition.

Say I was arrested and convicted of a crime in NH, but got it annulled and then stupidly moved to CT.  How should I respond to the question in the CT form?  I would say that I can legally say that I have not been arrested nor convicted.  What say you?  Thanks.!

Not a lawyer, but see the highlighted portions.

NH apparently law says that, if a charge is annulled, you can act as if it never happened.

That's my understanding of how it works in Ohio, at least: if a record is sealed, it's as if it never happened (except under some narrow circumstances).

YMMV

If you need an answer talk to an attorney licensed in the state you require to know about.

There are 50 states.

And if the question is for a federal activity (security clearance) they have other sets of rules.What you may get past on a confidential or secret may gig you on a TS with compartments.