Usually your state will post a note on their refund tracking page to tell you how long processing times can be, but not all states do this. And mailed returns will generally take longer than e-filed ones, to be processed.

This is on Arkansas's Department of Finance and Administration website for you Refunds - How Long to Wait and it states "The average processing time for a complete paper tax return is up to 10 weeks from the time you mailed it. Please allow 10 weeks from the date you filed before calling our office concerning the processing of a refund."

But TurboTax has no way to find out how long it'll take your state to send your refund once they process the return. You would have to check that with the state directly since they handle all processing and refunds. Once you file successfully with TurboTax and the state accepts, they handle things from there on out.