We study high-energy neutrino production in collimated jets inside progenitors of gamma-ray bursts (GRBs) and supernovae, considering both collimation and internal shocks. We obtain simple, useful constraints, using the often overlooked point that shock acceleration of particles is ineffective at radiation-mediated shocks. Classical GRBs may be too powerful to produce high-energy neutrinos inside stars, which is consistent with IceCube nondetections. We find that ultralong GRBs avoid such constraints and detecting the TeV signal will support giant progenitors. Predictions for lowpower GRB classes including low-luminosity GRBs can be consistent with the astrophysical neutrino background that IceCube may detect, with a spectral steepening around PeV. The models can be tested with future GRB monitors.PACS numbers: 95.85. Ry, 97.60.Bw, 98.70.Rz Long gamma-ray bursts (GRBs) are believed to originate from relativistic jets launched at the death of massive stars. Associations with core-collapse supernovae (CCSNe) have provided strong evidence for the GRB-CCSN relationship [1]. But, there remain many important questions. What makes the GRB-CCSN connection? How universal is it? What is the central engine and progenitor of GRBs? How are jets launched and accelerated? Observationally, it is not easy to probe physics inside a star with photons until the jet breaks out and the photons leave the system. This is always the case if the jet is "chocked" rather than "successful" [2]; that is, the jet stalls inside the star, where the electromagnetic signal is unobservable. Such failed GRBs may be much more common than GRBs (whose true rate is ∼ 10 −3 of that of all CCSNe), and CCSNe driven by mildly relativistic jets may make up a few present of all CCSNe [3][4][5].Recent observations suggest interesting diversity in the GRB population. "Low-power GRBs" such as lowluminosity (LL) GRBs [3,6,7] and ultralong (UL) GRBs [8,9] have longer durations (∼ 10 3 -10 4 s) compared to that of classical long GRBs, suggesting different GRB classes and larger progenitors. While they were largely missed in previous observations, they are important for the total energy budget and the GRB-CCSN connection.Neutrinos and gravitational waves (GWs) can present special opportunities to address the above issues. In particular, IceCube is powerful enough to see high-energy (HE) neutrinos at 1 TeV [10] and has reported the first detections of cosmic PeV neutrinos [11]. Efficient HE neutrino production inside a star has been proposed assuming shock acceleration of cosmic rays (CRs) [12][13][14], and investigated by a lot of authors, since their detection allows us to study the GRB-CCSN connection [13,15], joint searches with GWs [16], neutrino mixing including the matter effect [17], the nature of GRB progenitors [18] and so on. However, IceCube has not detected neutrinos from GRBs, putting limits on this scenario as well as the classical prompt emission scenario [19,20]. It also constrains orphan neutrinos from a CCSN [21].In this work, we consider HE ne...