Are there notifications in BQE CORE for managers when time is not submitted or the staff is not completing their time cards?
You are looking for a functionality that notifies your employees about missing or incomplete time cards. This is possible using the Time Card Monitoring feature wherein you can let CORE send out reminders to your employees for incomplete time cards and for not submitting the time entries. Check CORE Help Center for details.
Your employees can also turn on the reminder Time entries to be submitted by me in User Settings > Reminders to help them remind to submit time.