It typically has a length of 64 bits, though the length may vary depending on the chain. The private key is used to create signatures in a blockchain transaction, which serves to prove ownership of the funds being paid for. It's the secret half of your address and is a string of 64 hexadecimal characters. If you write down your private key differently today than yesterday, you'll be accessing a different wallet. Never write your private key by hand.

The recursive circuit takes 64 different proofs (π0, ..., π63), hashes (h0, ..., h63), equities (e0, ..., e63), and debts (d0, ..., d63) from the lower-layer circuits as inputs. It combines these inputs and produces 3 outputs: a new hash (hrecursive), total equity (erecursive), and total debt (drecursive).