An Arkansas contractor taking work in Kentucky 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 Kentucky's answer to each, with the agency's own link.
Licensing
Department of Housing, Buildings and Construction (Public Protection Cabinet)
No statewide general-contractor license; GC requirements are local. Housing, Buildings and Construction licenses electricians, plumbers, and HVAC contractors and journeymen statewide (HVAC licensed since 1995, KRS 198B.650-689), plus boiler and sprinkler and fire-alarm contractors.
Workers compensation
Kentucky Department of Workers' Claims, Education and Labor Cabinet
Employers with one or more employees must obtain and maintain workers' compensation coverage or Commissioner-approved self-insurance; part-time, temporary, and family-member employees all count.
https://elc.ky.gov/Workers-Compensation/Pages/Employer-Frequently-Asked-Questions.aspx
Business registration
Kentucky Secretary of State (filings via Kentucky Business One Stop portal)
New hire reporting
Most states require newly hired employees to be reported within a short window of their start date.
Kansas · All fifty states · Louisiana
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.