die Hoheit
sovereignty
The highest, sovereign power of a state over its territory and its affairs; sovereignty.
Der Staat besitzt die alleinige Hoheit über die Gesetzgebung.
The state has sole sovereignty over legislation.

sovereignty / highness
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sovereignty
The highest, sovereign power of a state over its territory and its affairs; sovereignty.
Der Staat besitzt die alleinige Hoheit über die Gesetzgebung.
The state has sole sovereignty over legislation.

'Die Hoheit' (sovereignty / highness) is derived from the adjective 'hoch' (high). It describes the quality of being 'hoch' (high)—whether in relation to power, rank, or aura.
Die Hoheit des Amtes verlangt absolute Neutralität.
The dignity of the office demands absolute neutrality.
In the meaning 'Souveränität' (sovereignty), the word is often used with 'über' (+ accusative). Typical phrases are 'die Hoheit über etwas haben' (to have sovereignty over something) or 'die Hoheit ausüben' (to exercise sovereignty). It also forms compounds such as 'Lufthoheit' (air supremacy) or 'Datenhoheit' (data sovereignty).
Jeder Staat hat die Hoheit über sein eigenes Territorium.
Every state has sovereignty over its own territory.
As a form of address for nobles, 'Hoheit' (Highness) is almost always accompanied by a possessive pronoun such as 'Ihre' (Her/Your) or 'Seine' (His/Your). Often an adjective such as 'Königliche' (Royal) or 'Kaiserliche' (Imperial) also comes before it.
Seine Königliche Hoheit, der Prinz, wird um 15 Uhr erwartet.
His Royal Highness, the prince, is expected at 3 p.m.
In the meaning 'erhabene Ausstrahlung' (elevated aura), 'Hoheit' (majesty / dignity) is often used in a poetic or very descriptive context. It frequently describes the effect of nature, art, or architecture.
Die Wanderer bewunderten die stille Hoheit der Alpen.
The hikers admired the quiet majesty of the Alps.
rule / control
'Herrschaft' and 'Hoheit' can both have to do with supreme power over a territory. 'Herrschaft' more often describes that someone actually exercises power and control. 'Hoheit' means the highest legal power of a state over its territory and its own affairs. The word sounds more formal and is often used in political or legal contexts. Learners may confuse the words because both talk about power. Put simply: 'Herrschaft' emphasizes ruling, while 'Hoheit' emphasizes the right to decide independently.

Der Staat verteidigt seine Hoheit über das eigene Gebiet.
The state defends its sovereignty over its own territory.
Nach vielen Jahren endete die Herrschaft der alten Regierung.
After many years, the old government's rule came to an end.