Support, uphold, advocate, back, champion are comparable when they mean to favor actively and in some concrete manner a person or thing that meets opposition.
Support is general and comparatively colorless. One supports a candidate for election whether one merely votes for him or takes a leading part in his campaign; one supports a cause whether one merely announces one’s stand in favor of it or contributes money and time to furthering its interests.
Uphold carries an implication of keeping erect or from falling or breaking down; it is appropriately used in reference to something that already exists but is attacked or challenged.
Advocate implies vocal support either in speeches or in writings; often it connotes urging or pleading.
Back often implies strong support from the rear to be used whenever assistance is needed to prevent the failure of a person or of his ventures or efforts. Sometimes it connotes reserve forces or the use of force; sometimes it implies money reserves or the promise of financial assistance.
Often back derives its implications from its use in betting and suggests a willingness to hazard something and especially money on a person’s or thing’s chance for success.
Champion implies public defense of a person or thing believed to be unjustly attacked or too weak to advocate its own cause and may therefore connote distinction or gallantry in the one who champions.