An Arkansas contractor taking work in New Jersey 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 Jersey's answer to each, with the agency's own link.
Licensing
NJ Division of Consumer Affairs (home improvement contractor registration); NJ Department of Community Affairs (new home builder registration)
Home improvement contractors must register with the Division of Consumer Affairs before selling or performing work; municipalities cannot issue permits to unregistered contractors. New home builders register separately with the Department of Community Affairs before starting construction.
Workers compensation
NJ Department of Labor and Workforce Development, Division of Workers' Compensation
All New Jersey employers not covered by federal programs must have workers' compensation coverage or approved self-insurance, including any employee working for compensation.
https://www.nj.gov/labor/workerscompensation/employer-requirements/
Business registration
NJ Department of the Treasury, Division of Revenue and Enterprise Services
Tax registration
NJ Division of Revenue and Enterprise Services (NJ-REG); taxes administered by NJ Division of Taxation
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.