Since the boom of smart phones and location services, spatio-temporal data (i.e., user locations with timestamps) have become increasingly essential in many real-life applications. To ensure these data are faithfully extracted from the underlying loc tracking hardware and not altered by any malicious party or the user himself/herself, integrity assurance schemes such as digital signatures or message authentication codes (MAC) must be adopted. However, these conventional schemes disclose to the verifier the complete plaintext location and thus jeopardize users' privacy. Propose an integrity assurance scheme with minimum location disclosure. That is, the granule of the disclosed location is just small enough to prove the user is/has been to a certain place, and the verifier cannot learn anything beyond it. To this end, a new MAC scheme called Prefix-verifiable MAC (PMAC), based on which we design indexes and protocols to authenticate both spatial and spatio predicates. Security analysis and experimental results show our scheme is both secure and efficient for practical use.