Kathleen Rice was a political neophyte in 2005 when she upset a seemingly entrenched Nassau County district attorney. Now, after less than five years as the county’s chief prosecutor, Ms. Rice, 45, is seeking the state government’s top legal job.

But while Ms. Rice might be relatively new to the political scene, she insists that her nearly 20 years of experience as a prosecutor makes her uniquely qualified among the five Democrats vying for the job.