1. Nf3
Two mover's
I think you're right, ilmari - Qc4. Good puzzle
The queen needs to stay where it is - to stay on the f1-a6 diaganol to quard Qf1# and to stay on the 6th rank to prevent Nd6# or Nxh6#.
The rook needs to stay on the 5th rank to prevent Re5#. Any other rook move leaves a mating threat.
The bishop can't move to prevent Nxg3 and Ne7 (Bxe7+).
The b and d pawn can't move or else it'll block either the queen or the rooks lines that prevent mate
It's a very good zugzwang!
For the first puzzle, can't you just checkmate in one move with White Qf7? Srry, I'm sorta new to this..
white to move, mate in two!!