But it doesn't explicitly prevent other players from moving the cubes around. They just can't put them into the row of squares.
So, instead of establishing 8 rows above the row of squares (per Y-san), just establish two--the "home row" and the "indicator row".
First player (last in line), who knows the second player's number, puts his/her own cube in the indicator row, putting it in the column where the 2nd player's cube should go.
Remaining players (knowing the next player's number) come in, place their cube in the home row below the indicator cube, then move the indicator cube to the column where the next player's cube should go.
(In other words, the first player's cube is used as the indicator cube all the way through).
Last player places their cube in home row below the indicator cube, moves indicator cube to the empty column, then moves each cube into the row of squares.
Saves a little real estate in principle (though you'd think Y-san's approach would work in practice), but would start to look real good if there were 100 players rather than 8.