An Arkansas contractor taking work in New Hampshire deals with the same four questions everywhere: who licenses the trade, when a licence is actually required, when workers compensation kicks in, and where the business registers. Below is New Hampshire's answer to each, with the agency's own link.
Licensing
Office of Professional Licensure and Certification (OPLC)
No statewide general-contractor license. Individual trades are state-licensed under OPLC: electricians (Electricians' Board); plumbers, gas fitters, and oil-heat technicians (Mechanical Safety and Licensing Board).
Workers compensation
New Hampshire Department of Labor, Workers' Compensation Division
Every employer with one or more employees must carry workers' comp; up to three corporate officers or LLC members may be excluded from the count (RSA 281-A:2).
Business registration
New Hampshire Secretary of State, Corporation Division (NH QuickStart)
Tax registration
New Hampshire Department of Revenue Administration
No general sales tax; businesses register and file Business Profits Tax and Business Enterprise Tax via Granite Tax Connect.
New hire reporting
Most states require newly hired employees to be reported within a short window of their start date.
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Starting or running the business itself is covered in the contractor business guide. Arkansas licence records for Benton and Washington counties are in the contractor directory.