How can I use private APIs to block incoming calls in an iOS application?

Core Telephony doesn't support this. To my knowledge there is no way to do this with any known private APIs either.

Are you sure it does not? code examples on http://tech.ruimaninfo.com/?p=83 shows how to do such things. Core Telephony headers in SDK are not complete. Of course this means no app store this is my code fragment based on example I linked

if ([str1 isEqualToString:@"kCTCallIdentificationChangeNotification"])
{
    NSDictionary *info = (__bridge NSDictionary *)userInfo;
    CTCall2 *call = (__bridge CTCall *)[info objectForKey:@"kCTCall"];
    NSString *caller = CTCallCopyAddress(NULL, call);
    NSLog(@"Caller %@",caller);
    if ([caller isEqualToString:@"+1555665753"])
    {
       //disconnect this call
       CTCallDisconnect(call);

}

additional definitions needed:

typedef struct __CTCall CTCall;
extern NSString *CTCallCopyAddress(void*, CTCall *);
extern void CTCallDisconnect(CTCall*);

and you need to monitor telephony center's callback(see linked example) I tested this fragment on my iOS5 device