We construct a delegation scheme for verifying non-deterministic computations, with complexity proportional only to the non-deterministic space of the computation. Specifically, letting n denote the input length, we construct a delegation scheme for any language verifiable in non-deterministic time and space (T (n), S(n)) with communication complexity poly(S(n)), verifier runtime n · polylog(T (n)) + poly(S(n)), and prover runtime poly(T (n)). Our scheme consists of only two messages and has adaptive soundness, assuming the existence of a sub-exponentially secure private information retrieval (PIR) scheme, which can be instantiated under standard (albeit, sub-exponential) cryptographic assumptions, such as the sub-exponential LWE assumption. Specifically, the verifier publishes a (short) public key ahead of time, and this key can be used by any prover to non-interactively prove the correctness of any adaptively chosen non-deterministic computation. Such a scheme is referred to as a non-interactive delegation scheme. Our scheme is privately verifiable, where the verifier needs the corresponding secret key in order to verify proofs. Prior to our work, such results were known only in the Random Oracle Model, or under knowledge assumptions. Our results yield succinct non-interactive arguments based on sub-exponential LWE, for many natural languages believed to be outside of P.