No. You do not need to submit a written policy or demonstrate compliance to the state in any way unless a complaint is filed against you by an employee.
Oversight for this legislation is complaint driven and follows the protocol set forward for non-payment of wages. If an employee believes that they have not received paid leave that is due to them, they may submit a complaint to the department of labor, who will then reach out to the employer to investigate the complaint. While there are not specific guidelines or standards for record keeping, it is important that you institute a system for tracking the paid leave you provide to your employees.