harry potter - Why does Voldemort hate half-bloods?

We know that Voldemort is a half-blood. He also hates half-bloods, we can see this from Harry Potter and the Half Blood Prince (Movie). When he takes command he starts killing half-bloods as well.

What is the reason to hate half-bloods even if he was a half-blood?

Answer

There were no targeted killings of half-bloods when the Dark Lord gained control. When the Dark Lord took power, he didn’t have half-bloods killed - he had Mudbloods, wizards born into Muggle families, locked up and sometimes killed. Those who could prove they had one close wizarding relative were spared.

“Nevertheless, unless you can prove that you have at least one close wizarding relative, you are now deemed to have obtained your magical power illegally and must suffer the punishment.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)}}

This would of course include all half-bloods, as all half-bloods have at least one parent who’s a wizard. Proving this further, one wizard in the hearings determining blood status claimed he was a half-blood as an attempt to get himself freed.

‘No, no, I’m half-blood, I’m half-blood, I tell you! My father was a wizard, he was, look him up, Arkie Alderton, he’s a well-known broomstick designer, look him up, I tell you – get your hands off me, get your hands off –”

_{^{- Harry Potter and the Deathly Hallows, Chapter 13 (The Muggle-born Registration Commission)}}

If half-bloods were hated under the Dark Lord’s rule, this wouldn’t help, and would be useless saying. The Dark Lord didn’t have half-bloods killed for being half-bloods. He surely would kill a specific half-blood if they were in his way - he was willing to kill anyone if they were sufficiently in his way. However, it’s clear that they weren’t actively tracked down by his Ministry to be removed from wizarding society by any means. In the official Ministry position, wizards with no proven wizarding ancestry were said to have stolen their magic.

“Recent research undertaken by the Department of Mysteries reveals that magic can only be passed from person to person when wizards reproduce. Where no proven wizarding ancestry exists, therefore, the so-called Muggle-born is likely to have obtained magical power by theft or force.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)}}

Half-bloods do have wizarding ancestry, though it’s not pure wizarding ancestry. So, this wouldn’t apply to them. Half-bloods were never targeted. There were Ministry rewards for capturing Mudbloods and blood traitors (pure-bloods who didn’t value their blood status) but none mentioned for half-bloods.

“Snatchers,’ said Ron. ‘They’re everywhere, gangs trying to earn gold by rounding up Muggle-borns and blood traitors, there’s a reward from the Ministry for everyone captured. I was on my own and I look like I might be school age, they got really excited, thought I was a Muggle-born in hiding. I had to talk fast to get out of being dragged to the Ministry.”

_{^{- Harry Potter and the Deathly Hallows, Chapter 19 (The Silver Doe)}}

Hermione says it’s only Mudbloods the Death Eaters hate, and there aren’t enough pure-bloods for them all to be pure-bloods. So it’s likely that the Dark Lord allows half-bloods into the Death Eaters as a general rule, not an exception - which he wouldn’t if he hated them like he hated Mudbloods.

“The Death Eaters can’t all be pure-blood, there aren’t enough pure-blood wizards left,’ said Hermione stubbornly. ‘I expect most of them are half-bloods pretending to be pure. It’s only Muggle-borns they hate, they’d be quite happy to let you and Ron join up.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 12 (Silver and Opals)}}

Also, it’s further implied that half-bloods are allowed to be Death Eaters, as J.K. Rowling says Snape being a Death Eater is a hint to his ancestry, because Mudbloods are rarely allowed to be Death Eaters.

Apart from Harry, Snape is my favourite character because he is so complex and I just love him. Can he see the Thestrals, and if so, why? Also, is he a pure blood wizard?

Snape’s ancestry is hinted at. He was a Death Eater, so clearly he is no Muggle born, because Muggle borns are not allowed to be Death Eaters, except in rare circumstances. You have some information about his ancestry there.
_{^{- Edinburgh Book Festival (August 15, 2004)}}

This implies that his ancestry isn’t unusual among the Death Eaters, further evidence that the Dark Lord only hates Mudbloods, as he wouldn’t allow half-bloods to join the Death Eaters if he hated them.