Following is a look at how that legislative power, the first held by a far-right party in Germany since World War Two, could work and what risks it could pose.
WHY IS A BLOCKING MINORITY PART OF THE LAW?
In Germany, some decisions of great consequence like changing the constitution or making appointments to bodies key to the functioning of democracy and the rule of law require a two-thirds majority.
Conversely, any one party that can marshal a third of votes in parliament - representing a significant portion of public opinion - can wield a blocking minority.
The two-thirds or super majority, a rule inspired in part by the Nazis' seizure of dictatorial power after winning the 1933 election, was designed to stop any one party capturing the state and transforming it. Sweeping change should be achievable only through agreement among multiple parties.
On the other hand, a blocking minority could be used by a party at odds with liberal democracy, like the AfD, to gum up and delegitimise the state, for example by snarling appointments to judicial and security bodies.
WHERE IS A TWO-THIRDS MAJORITY NEEDED?
In Thuringia, for appointments to the state Constitutional Court, the committee that appoints and confirms other judges, or the committee that oversees security services, and of the official in charge of the security agency which, among other things, monitors alleged far-right and extremist threats to democracy - including the Alternative for Germany (AfD).
WILL THE AfD BLOCK ALL NOMINATIONS?
The AfD has for years been accused of using legislative and judicial levers to bog down state institutions across Germany, maximising its clout even without formal legislative powers. It says it is merely making full use of its democratic rights.
The party is one of the most active sponsors of filings before the Federal Constitutional Court, packing out its docket with unmanageable case volumes.
WHEN DOES THE FIRST CRUNCH COME IN THURINGIA?
The first appointment that will require a two-thirds approval looms very soon: A substitute Constitutional Court judge who was elected to parliament on Sunday must be replaced.
HOW MUCH HARM COULD A BLOCKING MINORITY DO?
If no more constitutional court judges are appointed, every current judge's term will have expired by 2029. Lower-court judges in Thuringia are older than the national average - many have been office since German reunification in 1990 - so drawn-out attrition would leave courts ever less able to service their dockets, creating long waits for justice.
WHAT ARE THE CONSEQUENCES BEYOND THE JUDICIARY?
Surveys show the AfD thrives on the perception of many voters that a sclerotic and ineffective state needs radical change. Endless delays to rulings on everything from company filings and divorce papers to criminal cases could reinforce that perception, even if the AfD is itself partly to blame.
While small Thuringia has less than 3% of Germany's population, a spectacle of paralysis will be noticed far beyond its boundaries and could influence voters further afield ahead of next year's national election.
WHAT CAN BE DONE?
On the national level, government and opposition recently agreed to amend the constitution so a future populist party bent on political chaos to advance an authoritarian agenda would not be able to disable the Federal Constitutional Court, which is regarded as German democracy's ultimate guarantor.
It's too late for that in Thuringia: state politicians will have little choice but to try to strike deals, informal or otherwise, with the AfD, regardless of their stated refusal to work with the far rightists.
Legal scholars, like Erfurt University's Anna-Mira Brandau, have been looking into whether the federal government could delegate its own judges and courts to cases in the small state in Germany's formerly Communist east as an emergency stop-gap.
