An argument in favor of quora enforced during elections to enter parliament is that it guarantees more stability for the governing parties and thus for the government. I was wondering whether this holds true and if yes which is the minimum quorum for which this is true.
No electoral system, other than single party rule, can guarantee government stability.
The requirement of a minimum amount of support from voters to have a representative in parliament was an innovation traceable pretty much to post-war Germany, which in the wake of the failed Weimer Republic imposed a 5% of the vote threshold to get seats in parliament in order to avoid legitimatizing extremists like Neo-Nazis with the bully pulpit and power in the event of an even divide between parties that can't form a coalition with each other, that comes from having of seats in parliament.
Empirically, what a 5% requirement does is limit the number of viable political parties that will be able to gain representation in parliament to roughly 4 to 6 political parties, as opposed to the two political parties in any given region which a first past the post, single member district system naturally favors on one hand (which is effectively equivalent to a minimum requirement of something like 40% support in practice and 50%+ support when only two candidates are running), and as opposed to the 6 to 12ish number of political parties that a pure proportional representation system without a minimum popular vote threshold tends to generate (or course, even in a PR system there is always, at least, an approximately 1/number of seats in parliament percentage of the vote minimum of support, and more if the nation is broken up into several districts for PR vote allocation purposes).
The higher the minimum percentage of support you need in your electoral district you need to get a seat in parliament, the fewer political parties you will have and the more necessary it will be for political parties to form coalitions before the election. The lower the threshold of support you need to get a seat in parliament, the more political parties you will have and the more likely it is that coalitions will be made after the election, rather than before it.
Two examples of countries with close to pure PR systems without meaningful quotas that have long histories of unstable coalition governments are Italy and Israel.
For example, for most of the post-war history of Italy, every government has been a coalition of several of five more moderate parties who banded together to keep hardline communists on the left, and neo-fascist parties on the right out of government. This required government coalitions to span exceptionally broad ideological divides, while requiring near unanimous support on a continuous basis within government coalitions, which in turn made those coalitions very unstable producing a constant revolving door of new governments with most of the same players.
But, there is nothing magic about a particular percentage threshold like Germany's 5%. If you set a 10% threshold, you might get 3-4 parties. If you set a 3% threshold, you might get 5-8 viable parties. The coalition of parties necessary to form a government, in turn, will typically be about half, plus or minus, of the total number of parties with seats in parliament.*
Of course, the more parties that are necessary to form a coalition government, the less stable those governments will be, and it is especially hard when some of the parties are extremist parties that don't want to play by the ordinary political rules and may have little interest in actually governing.
Whether a quorum actually keeps extremists out of office depends largely upon how much support extremist parties have. In ordinary times, a 5%-10% threshold is going to keep the parties that get into office fairly moderate. But, a sharp economic downturn or scandal or charismatic leader can push support for extremist factions much higher than any reasonable quorum.
- Caveat: This general rule is subject to two main exceptions.
FIRST: The very first election under a new PR political regime typically attracts a huge number of parties and gets many elected as coalitions haven't had time to form yet. Usually, the system mostly settles down with a more stable long term set of parties by the second election under the new system, and hits a long term equilibrium by the third election that lasts until something catastrophically disrupts one or more existing parties leaving a vacuum (such as in one of the recent Greek elections).
SECOND: If there is strong nationalist division within a country that has a unitary parliament, the number of parties will increase as each cultural nation with a regional identity develops its own political ecology (e.g. Scotland and Northern Ireland in the U.K., or Quebec, Western Canada and Eastern Canada in Canada, or Kurdistan, Shiite regions and Sunni regions in Iraq).