This paper explores the preparation of matrix-product states (MPS) via fusion, focusing on the constraints and extensions of this method. The authors investigate the deterministic preparation of MPS in constant depth using measurements and classical communication to fuse smaller states into larger ones. They establish that for MPS to be prepared via fusion, they must have a flat entanglement spectrum. This constraint is demonstrated using the AKLT state, which is a symmetry-protected topological (SPT) phase and a resource for quantum computation. The authors also introduce a family of states protected by non-onsite symmetries, which lie beyond the scope of ordinary MPS fusion. These states can be prepared in constant depth using a broader class of measurement-assisted protocols, termed SIMPS fusion. The results show that SIMPS fusion can significantly reduce resource overhead compared to MPS fusion. The paper also discusses the implications of these findings for the preparation of many-body quantum states in quantum computers, highlighting the importance of understanding the limitations and potential extensions of MPS fusion. The authors conclude that SIMPS fusion provides a more general approach to preparing states with non-trivial symmetries and non-flat entanglement spectra, enabling the efficient preparation of many-body states belonging to new phases of matter.This paper explores the preparation of matrix-product states (MPS) via fusion, focusing on the constraints and extensions of this method. The authors investigate the deterministic preparation of MPS in constant depth using measurements and classical communication to fuse smaller states into larger ones. They establish that for MPS to be prepared via fusion, they must have a flat entanglement spectrum. This constraint is demonstrated using the AKLT state, which is a symmetry-protected topological (SPT) phase and a resource for quantum computation. The authors also introduce a family of states protected by non-onsite symmetries, which lie beyond the scope of ordinary MPS fusion. These states can be prepared in constant depth using a broader class of measurement-assisted protocols, termed SIMPS fusion. The results show that SIMPS fusion can significantly reduce resource overhead compared to MPS fusion. The paper also discusses the implications of these findings for the preparation of many-body quantum states in quantum computers, highlighting the importance of understanding the limitations and potential extensions of MPS fusion. The authors conclude that SIMPS fusion provides a more general approach to preparing states with non-trivial symmetries and non-flat entanglement spectra, enabling the efficient preparation of many-body states belonging to new phases of matter.