The phrase "give a leg up" is typically used to signify helping someone to advance or succeed by giving them a boost or advantage. Antonyms for this phrase could include phrases such as "knock down" or "pull down," which mean hindering progress or discouraging someone's success. Other antonyms might include phrases like "ignore" or "neglect," which suggest not providing any assistance whatsoever. Conversely, synonyms for "give a leg up" might include phrases like "support," "aid," or "assist," all of which suggest actively helping someone to achieve their goals.