Explore a groundbreaking paper on post-quantum cryptography and transparent recursive proofs presented at Eurocrypt 2020. Delve into the foundations of recursive composition of SNARKs, post-quantum recursion theorems, and holographic proofs. Learn about the Fractal system's evaluation as both a standalone and recursive SNARK, and discover the implications of preprocessing for efficient recursion. Examine the concept of post-quantum preprocessing SNARKs and their connection to holographic proofs. Investigate the potential for efficient holography and linear-size encoding of sparse matrices in this cutting-edge cryptographic research.