This question is about a puzzle from the book of Raymond Smullyan: The Lady or the Tiger? The description of the puzzle starts in Chapter 8, p. 103, and here is the important part (copy-pasted from Bruce Ediger's webpage):


Property Q: For any combination x, the combination QxQ is specially related to x.


Property L: If x is specially related to y, then Lx is specially related to Qy.


Property V (the reversal property: If x is specially related to y, then Vx is specially related to the reverse of y.


Property R (the repetition property: If x is specially related to y, then Rx is specially related to yy.


Property Sp: If x is specially related to y, then if x jams the lock, y is neutral, and if x is neutral, then y jams the lock.


Smullyan provides a key of length 10 on p. 163, however, this is not the shortest, I know a key of length 8. Is this the shortest?
Spoiler alert: You can find this key in my comment to the accepted answer.)



Shortest Key for the Monte Carlo Lock of Smullyan
Edited in recognition of closed-mindedness.


~~My brute force search shows no keys shorter than 10. Here are the only keys of length 10 and 11 respectively:~~


RVLVQRVLVQ
VRLVQVRLVQ


VLRVQVLRVQQ
VLVRQVLVRQQ


Curiously, there are no keys of length 12. 


The only word of length 7 that does not crash under iteration is RRQRRQQ, and it evolves unboundedly. There are 74 words of length 8 that grow to over 30 letters, I think none of them cycles, and there are two eventually cycling words, one you gave and the other its pair RQVRLVQQ. The first time an odd period greater than 1 appears is at length 12, these are the originating words (all end up with period 3):


RQQVLLRLVQQQ


RLQVLLRLVQQQ


RQLVLLLRVQQQ


RQVLLLVLRQQQ


RQLVLLLVRQQQ


RQLVLLLRVLQQ


RQVLLLVLRLQQ


RQLVLLLVRLQQ


RQLLVLLRLVQQ


RQVLLLVQQRQQ


RQVLLLVLQRQQ


RQLVLLLVQRQQ


