The MPC in the Head (MPCitH) paradigm has recently led to significant improvements for signatures in the code-based setting. In this paper we consider some modifications to a recent twist of MPCitH, called Hypercube-MPCitH, that in the code-based setting provides the currently best known signature sizes. By compressing the Hypercube-MPCitH five-round code-based identification scheme into three-rounds we obtain two main benefits. On the one hand, it allows us to further develop recent techniques to provide a tight security proof in the quantum-accessible random oracle model (QROM), avoiding the catastrophic reduction losses incurred using generic QROM-results for Fiat-Shamir. On the other hand, we can reduce the already low-cost online part of the signature even further. In addition, we propose the use of proof-of-work techniques that allow to reduce the signature size. On the technical side, we develop generalizations of several QROM proof techniques and introduce a variant of the recently proposed extractable QROM.