Here is an Arizona news story about the Libertarian Party’s new ballot access lawsuit, filed April 12. The story quotes several Republican legislators defending the 2015 law change that made it very difficult for Libertarians to get on the Libertarian primary ballot. They say the old law, which only required a few hundred signatures of Libertarian Party members for a candidate to get on a Libertarian primary ballot, was too easy because all candidates need to show a significant level of voter support.
The problem with that argument is that the state has already conceded the Libertarian Party itself has a modicum of support (because it constantly remains ballot-qualified, because its registration is always above two-thirds of 1% of the state total). Therefore, anyone who wins the Libertarian primary does have a modicum of support, because the party backs that person and the party has already shown it has support.
Most states don’t require any signatures for a candidate to get on a partisan primary ballot. Instead they use filing fees.
Furthermore, the purpose of petitions (whether for a general election ballot or a primary ballot) is to keep the ballot from being too crowded. But the old law already kept the Libertarian primary ballot from being crowded; it was very rare for more than one Libertarian to qualify for any particular office. So already the Libertarian primary ballot was not crowded, and there was no legitimate reason for the legislature to increase the burden.