As the "agentic web" takes shape-billions of AI agents (often LLM-powered) autonomously transacting and collaborating-trust shifts from human oversight to protocol design. In 2025, several inter-agent protocols crystallized this shift, including Google's Agent-to-Agent (A2A), Agent Payments Protocol (AP2), and Ethereum's ERC-8004 "Trustless Agents," yet their underlying trust assumptions remain under-examined. This paper presents a comparative study of trust models in inter-agent protocol design: Brief (self- or third-party verifiable claims), Claim (self-proclaimed capabilities and identity, e.g. AgentCard), Proof (cryptographic verification, including zero-knowledge proofs and trusted execution environment attestations), Stake (bonded collateral with slashing and insurance), Reputation (crowd feedback and graph-based trust signals), and Constraint (sandboxing and capability bounding). For each, we analyze assumptions, attack surfaces, and design trade-offs, with particular emphasis on LLM-specific fragilities-prompt injection, sycophancy/nudge-susceptibility, hallucination, deception, and misalignment-that render purely reputational or claim-only approaches brittle. Our findings indicate no single mechanism suffices. We argue for trustless-by-default architectures anchored in Proof and Stake to gate high-impact actions, augmented by Brief for identity and discovery and Reputation overlays for flexibility and social signals. We comparatively evaluate A2A, AP2, ERC-8004 and related historical variations in academic research under metrics spanning security, privacy, latency/cost, and social robustness (Sybil/collusion/whitewashing resistance). We conclude with hybrid trust model recommendations that mitigate reputation gaming and misinformed LLM behavior, and we distill actionable design guidelines for safer, interoperable, and scalable agent economies.