What branch of government is constitutionally responsible for establishing the lower courts

Legislature

Explanation:

Legislature is a branch of government responsible for making laws. Apart from law making laws, they can also enact bills to set up institutions like lower courts for efficient discharge of justice. For example, In the United States, the Congress is the legislative body charged with that responsibility of law making. But in addition and according to the federal constitution of the United States, the Congress also has power to set up lower courts.

So it is the legislative arm of government that is constitutionally responsible for establishing the lower courts.