Processing times vary from state to state; only your state can give you an exact date. Some states have noted they may take 90 days or longer to process returns and issue refunds. If you have successfully filed and the returns show as accepted in TurboTax, you will need to check your state's website using the Where's My Refund link below, to track the status of your refund.
They will provide the most up to date status for you there. The state does not send any updates about your return or refund to TurboTax after they accept your return.
NOTE: If you track status and it shows as "no record;" that just means they have not started processing your return. Until they start reviewing/processing it, there will be no record. Just keep an eye out on the status page for updates.
How do I track my Minnesota tax refund?