I am designing a protocol where a user A sends a time-locked deposit to user that if the user B within time t provide a proof that he exists in a location x (this may be obtained by third party like Google), user B can get the deposit of A. Otherwise, A can get his money back. Any help of how to do that using cryptography or two-way secure computations?
As this is an Ethereum forum, I assume you are interested in creating a contract which implements such functionality.
What you need is a contract which:
1) Accepts Ethers from user A
2) Upon receiving proofs of x from an oracle (oracle can get the data from for example Google) checks the timestamp. If current time is greater than t the contract forwards the stored Ethers to user B. Otherwise it probably does nothing.
3) According to some rules if user B can't provide requires proofs the contract sends the Ethers back to A.
This is a basic approach. To make it a bit more secure, consider applying the withdrawal pattern (https://firstname.lastname@example.org/why-use-the-withdrawal-pattern-d5255921ca2a) instead of sending Ethers.