New Hampshire car insurance laws are unusual in that car insurance is not required on New Hampshire roadways -- drivers only need to demonstrate the ability to pay for damages they cause in an accident. If you have an "at fault" accident without having insurance coverage, you will be required to post a bond or cash equal to the amount of damage you caused in that accident.
If at fault in an accident in this state you must also satisfy the New Hampshire Financial Responsibility requirements to continue to operate a vehicle in New Hampshire. These minimum limits are 25/50/25 (Bodily Injury $25,000 per person; $50,000, 2 or more persons; and Property Damage $25,000). A $75,000 single limit policy will also satisfy the minimum requirement.
Even though your home state may not require auto insurance coverages typically other states do require you to have auto insurance to operate a vehicle on their roadways. Most states require either insurance or other financial responsibility be placed on your car that would cover an accident you may cause out of state (conform to the state laws of where you are driving) and this would be good enough to show as insurance coverage to not be cited for driving without insurance.