State capital describes a political role.
The state parliament meets at Alter Markt in Potsdam. The State Chancellery supports the state government and is also based in Potsdam. These institutions help connect the city with state politics in your memory.
Potsdam is also a city with its own local administration. State politics and city politics are different levels, even when they take place in the same location. News from Potsdam may therefore concern the state parliament, state government or city itself.
The other three answers
The other cities belong to Brandenburg, but do not have the capital’s role.
Cottbus.
Cottbus is an independent city in Brandenburg. The state capital, however, is Potsdam.
Brandenburg.
The shortened answer Brandenburg here refers to the city of Brandenburg an der Havel. Do not confuse it with the whole federal state; it is not the capital.
Frankfurt/Oder.
Frankfurt/Oder refers to Frankfurt (Oder), another independent city in Brandenburg. It is not the state capital either.
Being close to Berlin does not change the state relationship.
Potsdam borders Berlin. Berlin is a separate federal state and Germany’s national capital. Sharing a capital region does not make Berlin Brandenburg’s state capital.
Independent city means that a city does not belong to a district and performs the corresponding functions itself. It does not automatically mean capital. All four answer cities are independent cities; only Potsdam is also Brandenburg’s state capital.
Brandenburg · Geography · Brandenburg state government ↗Brandenburg state → capital Potsdam.
Brandenburg an der Havel is a different city, not the capital.
Three words for this question
- die Landeshauptstadtstate capital
- The capital of a federal state.
- die StaatskanzleiState Chancellery
- An institution supporting and coordinating the work of the state government.
- kreisfreiindependent of a district
- A city belongs to no district and also performs the corresponding district functions.