how long does the cycle length have to be in stream ciphers? could someone tell me how long the cycle length must be in stream ciphers, so a security exists. how to compare the cycle length to the state space?

I investigate the algorithm spritz and I have the attainability of the states under investigation with key lengths of 1 - 10 and n = 6. now I have the values ​​but I still do not know how to interpret them. the maximum cycle length for N = 6 is 660.390 and the smallest 12. But how should I evaluate this with regard to the safety? I hope I have described it somewhat better.
There's a bioinfomatics paper that might be of interest.  Good Practice in (Pseudo) Random Number Generation for Bioinformatics Applications suggests some values to the maximum run length of a PRNG.  Not strictly crypto, but I've not seen anything else.

In summary, it says that Knuth recommends a max. output of states /1000. However there is also a much more conservative limit of (states)^(1/3).

I'm not sure of how you exactly determined your baby Spritz cycles, but for comparison consider the grown up version with the full 256 byte permutation and assume that all states are achieved.  (This may not be possible due to the creation of weak states.) The conservative estimate would suggest that you could output ~10^167 bits of key material.

For N = 6, I estimate that you couldn't get a single safe bit from it if you follow the guidance above.